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TEANSACTLIONS

OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

ESTABLISHED Novemser 15, 1819.

VOLUME THE FOURTH.

CAMBRIDGE:

PRINTED BY J, SMITH, PRINTER TO THE UNIVERSITY;

AND SOLD BY J. & J. DEIGHTON, AND T. STEVENSON, CAMBRIDGE ;
AND T. CADELL, STRAND, LONDON,

M.DCCC.XXXIII.

‘ce
; ©
.

CONTENTS OF THE FOURTH VOLUME.

Parr I.
PAGE
Ne. Primriz Faune et Flore Madere et Portus Sancti; sive Species quedam
Nove vel hactenus minus rite cognite Animalium et Plantarwm in his Insulis
degentium breviter descripte: by R. T. Lowk, Esq.............0ee cee eee eee I

II. On the General Equation of Curves of the Second Degree: by Auaustus
De Morean, Esq., Professor in the University of London................-. 71

III. On the Nature of the Light in the two Rays produced by the Double Refrac-
Honvoje Quartz: (oy) Professor AURY2h\ +e sea teioie ae sie!e silesloctan(l torte ne «are 79

IV. On the Resolution of Algebraical Equations: by the Rey. R. Murpuy......... 125

V. Mathematical Exposition of some of the Leading Doctrines in Mr. Ricardo's
*« Principles of Political Economy and Taxation: by the Rev. W. WHEWELL 155

Addition to a Paper “Qu the Nature of the Light in the Fwo Rays produced
by the Double Refraction of Quartz:” by Professor AIRY....--....-.++++5- 199
Part II.

VI. Desoription of Chiasognathus Grantii, a new Lucanideous Insect forming the
type of an undescribed Genus, together with some brief Remarks upon its
Structure and Affinities. In a Letter addressed to one of the Secretaries: by
Jn LP RIESE, TDG 8) ass oor docano BuO DACH BO OADCOponOnUGouOGCODDDEOUCDaOo Aut:

VII. A case of Human Monstrosity, with a Commentary: by the Rev. W. Crarx, M.D. 219

VIII. On the Examination of a Hybrid Digitalis: by the Rev. Professor HENstow...... 257

IX. Ona Remarkable Modification of Nemton’s Rings: by Professor A1ry........+..- 279

X. A Monograph on the British species of Cyclas and Pisidium: by the Rey. L. JENYNS 289

XI. On anew Analyzer, and its Use in Experiments of Polarization: by Professor Airy 313

CONTENTS.
Part III.
PAGE
On the Mechanism of the Larynx: by the Rev. R. WILLIS......--++ 0 seer ene 323
On the Inverse Method of Definite Integrals, with Physical Applications: by
he REV. MR AIMED SRB EE Maeeneye tote tere ore isaice] ote ancterseta latest are nia, s'sie s[olnfors\sinja\e\s¥rerate 353

On the Phenomena of Nenton's Rings when formed between two transparent
Substances of different refractive Powers: by Professor A1ry .........-...-. 409

Description of a Machine for resolving by Inspection certain important Forms
of Transcendental Equations: by Sir J. F. W. Herscuer, K.G.H., &c..... 425

Pawn Ss AC TPONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

Vou. IV. Parr I.

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9 a —— whe ie Lai

I. Primitie Faune et Flore Madere et Portus Sancti ;
sve Species quedam Nove vel hactenus minus rite
cognite Animalium et Plantarum in his Insulis de-
gentium breviter descripte.

Curante Ric. Too. LOWE, A.M.

COLL: CHR. CANT. ET NUPER AB EADEM UNIVERSITATE BACC. PERIGR.
[Read Nov. 15, 1830.]

ScrenTi# naturalis fautoribus haud quodammodo inutile
futurum speravi, (obstantibus multis quominis Prodromum meum
Faune et Flore Maderensis jam jamque edere possim), si spe-
cilerum novarum vel hactents mints cognitarum selecta quadam
characteribus brevibus statim exprimere curem: adjectis annotati-
unculis quibusdam, supervacaneis autem omnibus excisis, prout
brevitati vel maximé consulenti oportet.

Opusculum itaque de quo nune agitur, quasi Prodromus
Prodromi, characteribus constat specificis rerum sine ordine et
omnin6o ex arbitrio selectarum. Multa etiam nova omissa: plu-
rima incerta alteri diei studioque accuratiori relicta sunt. Hee
preecipue de re Entomologica et Ichthyologicé praemonenda velim :
quippe ope in Insectis describendis, qué prorsus confisus eram,
a morte amici C. Heineken M.D., scrutatoris vel oculatissimi,
orbatus sum; itaque rem omnem (in tanta specierum novarum
difficultate, entomologico vel peritissimo rem momenti non levis),
radicitis suscipere insolitus inusitatusque cogor. De Molluscis,
quim omnium terrestrium hactenis 4 me repertaruin catalogum
completum exhibere curavi, marinas feré omnes in preesens

Vol. IV. Part I. A

omisi. De plantis Acotyledoneis veris (Cellularibus), de Crusta-
ceis, Zoophytis, &c. idem est pradicandum. Plurima denique
dubia vel nondum satis explorata in partibus omnibus consulté
omisl.

Quod si in speciebus tam zoologicis quam botanicis novis rité
definiendis, hac aliquantulum valeant, fructum inde percipere,
haud parvum, spero; ex emendationibus, scilicet, amicis consiliis-
que omnium scientiz naturalis fautorum. Rei quidem herbarize
cultoribus maximé precor ut mihi quasi PoravopAw potiis quam
Boravocopy indulgeant. Ommnes denique rogo, quod si in aliis
corrigendis nimis aliquando videar audax, arctissimo tamen veri-
tatis natureque studio me semper pro viribus eniti credant:
omni petulantid mutationisque vane proclivitate ab animo longé
amota.

Amicis tantis tanta debenti, singulos enumerare, sua cuique
ascribenti, locus jam non adest: nec tamen omnes silentio pra-
termittere possum. Cl. Rob. Brown, adjuvante J. I. Bennett
arm’., summa humanitate ac benevolentia, plantarum Maderen-
sium 4a Masson aliisque lectarum, in Herbario Banksiano conser-
vatarum, necnon Manuscriptorum ipsius Solandri, copiam fecit.
Synonymiam certissimam Helicum a cl. Sowerby, Wood, &c.
descriptarum, ex autopsid speciminum ipsorum, partim in Museo
Britannico repostorum, benevolentize cl'. Children, Gray, et G. B.
Sowerby ipsius debeo. Amicus M. J. Berkeley characteres plan-
tarum plurium novarum ab ipso in Britannid cultarum, propriis
observationibus confirmavit, et iconibus accuratissimis illustravit.
Cl. Hooker litteris et amicitiz que quantaque non debeo! So-
cietati denique huic illustrissimz, si quid utilitatis, si quid com-
modi Scientiz fautores ex his laboribus (vel minimum) haurire
possint, gratias liceat omnium simul cum meis offerre: quippe
que primim inceptis nostris eximid liberalitate afflavit, eorum
primitias illa jam pritis inde percipere debet.

Pars [,

ie aN i a:

VASCULARES, D.C.

Cuass: II: MONOCOTYLEDONES,
A. CRYPTOGAME.

Orp. I. FILICES.

Gren. ASPIDIUM, RAR. Br.

1. Aspiprum falcinellum, Sw.

A. fronde pinnata: foliis falcato-ensiformibus, acuminatis, arguté
serrulatis, coriaceis, rigidis, petiolatis, basi surstm obtusé auriculatis:
soris biserialibus, approximatis, distinctis: indusiorum margine stellatim
dentato: stipite squamoso rhachique hirsutis.

Aspidium falcinellum, Sw. Syn. Fil. pp. 46, 243.

Hab. in cacuminibus montium. Madere; locis potissimim umbrosis,
frigidis gaudens.

Cum 4. auriculato, trapexoide, Sw. &c. species a plurimis confusa.
Indusia magna, orbiculata, pulcherrima. Sori conferti, aquidistantes,
haud confluentes. Nomen synonymiamque cl. Swartzii hane speciem
olim describentis, pro novo in MSS. imposito, retinendum curavi: mo-
nentibus primtim amicis cl. Rob. Brown et J. I. Bennett armig*; quo-
rum indicatione rem totam, ex autopsia speciminum in Herbario
Banksiano asservatorum, luce clariorem demonstratam habeo. Swartzio
A. falcinelli sui locus natalis plané latuit: quaproptér species Swart-
ziana omnibus valdé dubia; nec minis quam stirps maderensis cum
aliis quibusdam 4 plurimis confusa hucusqie visa est.

6 Mr. Lowe on the New Plants and Land Mollusca
2. <Aspidium frondosum, Prodr. MS.

A. fronde triangulari, tripinnata, subtts hirto-paleacea, ramosa,
ramis inferioribus adscendentibus: foliolis (tertii ordinis) oblongis ova-
tisque, acutis, inciso-dentatis, basi pinnatifidis, laciniis imis obtusis, den-
ticulatis, ima exteriore (superiore) majore; superioribus acutis, sub-mu-
cronatis: soris in foliola biserialibus, demtiim confluentibus: indusiis
confertis, pellucidis, adpressissimis, planis, demim marginibus reflexis:
rhachibus stipiteque basi hirsutissimo pallidis, paleaceo-hirtis.

Polypodium frondosum, Sol. MLSS’!

Hab. in sylvis rupibusque siccis umbrosis Madere: sterile haud in-
frequens; cum fructu rariss. :

Frondes nitide, lucide: fertiles supra granulate s. tuberculate.
Indusia in vivo squamiformia, orbiculata, maxima, membranacea, albida,

elegantia. Longit. cum stipite 2—4 pedes; Lat. 9 poll. ad 14 feré
pedum. Stipes 1—2 pedalis.

3. Aspidium? drepanum, Sv.

A? fronde lanceolata, acuminata, bipinnata: pinnis acuminatis, cur-

' vyato-adscendentibus, remotiusculis: foliolis angustis, acuminatis, sub-fal-

catis, arguté inciso-serrulatis; inferioribus sub-oppositis; infimo superiore
valdé elongato, rhachi parallelo; summis confluentibus: soris minutis,
confertis, distinctis, biseriatis: rhachibus stipitibusque densé paleaceis.

Aspidium drepanum, Sw. Syn. Fil. pp. 54, 255.

Hab. in convallium umbrosis Madere; rariss.

Frondes rigidiuscule, 2—3 pedales; pinnis 5—6 pollices, foliolis 1
pollicem feré longis. Frondes_ steriles dissimillime, pinnis basi tan-
tium pinnatis, foliolisque oblongis, multd latioribus, foliosis.

Indusia non vidi. Ob fructum in omnibus exemplaribus nimis
maturum, de genere dubitandum; sc. an species Polypodiis, Nephrodivs,
an Aspidiis rité consocianda. Indusia tamen, ut in <Aspidio caduco,
Hook. et Grev. Ic. t. 171. minutissima, vel citd caduca esse suspicor.

of Madera and Porto Santo. 7

Gen. NEPHRODIUM, R. Br.
4. Nephrodium focenisecii, Prodr. MS.

N. fronde triangulari vel ovata, 3—4 pinnatifida, utrinque glabra:
laciniis (tertii 4-tique ordinis) oblongis, obtusis; ultimis incisis, mucro-
nato-serratis; omnium inferioribus exterioribus internis Oppositis majo-
ribus: soris numerosis distinctis: indusiis primd semiovatis vel renifor-
mibus, demtm orbiculatis, emarginatis: stipite breviusculo, basi spar-
sim sub-paleaceo, fusco, superné rhachique pallidis.

a. dlatum; fronde 4—pinnatifida; pinnis inferioribus (1™ 2*° ordinis)
triangularibus vel ovatis, externis interioribus oppositis valdé majo-
ribus: pari infimo pinnarum (1™ ordinis) basi deorstim ramoso;
pinnula (2 ordinis) potissimim 1™ (aliquando etiam 2%) inferiore
s. exteriore deorstim producta.

Hab. in sylvis Vaceinii padifolii, Sm:, Madere; ubique vulgatissima.

B. productum; fronde tripinnatifidé, paulld magis elongata: pinnis om-
nibus oblongis; externis internis oppositis vix majoribus: lacini-
arum ultimarum dentibus sub-aristatis.

Hab. in umbrosis humidioribus Mader; rariss:

8. Status potits prioris (a), é loco obscuriore, defectu luminis, &c.
quam varietas videtur.

Frons in utraque varietate nana, 1—1} pedes (una cum stipite)
longa, feré pedalis; 6—8 pollices lata: stipite vix dimidium totius lon-
gitudinis equante. In utraque odor idem gratissimus, foenum novum
redolens, constans.

Species Aspidio dilatato et spinuloso Auct. certe proxima; et cum
illis forsan, in unam speciem (ut ab amiciss. cl. Hookero) consociatis,
olim conjungenda. Sed distingui posse credo, figura frondis abbrevi-
ata, deltoided; stipite breviore, minus (sc. basi tanttim) paleaceo; pin-
nulis angustioribus; odore. His adde frondem magis decompositam :
quamvis enim rard, sc. in 8, certé mints quam in a, decomposita, in

8 Mr. Lowe on the New Plants and Land Mollusca

utroque tamen statu saltém sub-ripinnata, et longé frequentitis, sc. in
a, statu normali, sub-guadripinnata*, Hee omnia, cum aliis characteri-
bus supra indicatis, millibus exemplaribus stabilita sunt; et in planta a,
aded per totam Insulam pervulgata, constantia, nec in tanta differentia
loci coelique (8. enim potitss monstrosa) variantia inveni.

Gen. ASPLENIUM, Linn. Spr. &c.

5. Asplenium anceps, Sol. ILS.

A. fronde pinnata, lineari-lanceolata: pinnis distinctis, sub-petio-
latis, oblongis, obtusis, apice sub-crenulatis, basi abruptis inferioribusque
sursim acuté auriculatis: soris biseriatis, obliquis, distinctis: rhachi
stipiteque nitidis, fuscis, trigonis, alato-marginatis.

Asplenium anceps, Sol. MLSS'!

Hab. in Madera, vulgaris; Asplenii Trichomanis locum tenens.

Gen. GYMNOGRAMMA, Desv. Hook.

6. Gymnogramma Lovei, Hook. et Grev.

G. fronde pinnata, utrinque hirsuta: pinnis oblongis, acuminatis,
pinnatifidis ; summis confluentibus: laciniis ovalibus oblongisve, obtu-
sissimis, integerrimis: stipite sparsim squamoso rhachique hirsutis.

Gymnogr: Lovei, Hook. et Grev. Ic. Fil. t. 89!

Gymnogr: Totta, “Schlechtend: (Polypod: tottum, Willd.) Spr.
Syst. IV. 1. p. 38. No. 6?

Acrostichum pilosum, So/. MSS. et Herb. Banks!

Hab. in Madere umbrosis humidioribus.

Pinne infime brevitér petiolate; mediz sessiles; summz conflu-

entes: inferiores remotiores.

* Ob lacinias omnium ordinum superné confluentes, rectils 3—4 pinnatifide frondes
scribuntur. Tamen ob tenuitatem suam 8—4 pinnate apparent.

of Madera and Porto Santo. 9

Orv. Il. LYCOPODIE.
Gen. LYCOPODIUM, Linn., Spr.

7. Lycopodium suberectum, Prodr. MS.

L. (capsulis axillaribus): caule erecto, dichotomo; basi incurvo,
decumbente: ramis fastigiatis: foliis squarrosis, 11—12 fariam imbri-
catis, lineari-lanceolatis, acuminatis, rigidis, sub-pungentibus; inferioribus
reflexis; superioribus erecto-patentibus.

Hab. in salebrosis fissurisque rupium sylvarum Madere.

Rami conferti, cespitosi, strictissimi, recti, crassitie digiti minim,
19 ad 16 pollices alti. Species L. Selagini proxima; sed preter alia,
longé major. Inter hane et ZL. avillare Roxb. et L. Saururum Lam.
et Bory, quodammodd intermedia: 4 L. Selagine tamen habitu _potis-
simtm distincta. Characteribus L. rigido Sw. potitis accedit, sed ha-
bitu omnino alieno. ;

B. PHANEROGAM .

Orb. ITI. GRAMINEZ.
Gen. AIRA, Sm.

8. Aira argentea, Prodr. MS.

A. cespitosa: panicule coarctate, apice nutantis, ramis verticillatis,
scabris: flosculis calycem equantibus, basi pilosis: aristé subdorsali, sc.
ad imum feré valve nascente, recta, flosculos duplo excedente: foliis
conduplicatis, filiformibus, compressis.

Hab. in Madere sylvis salebrosis.

Gren. FESTUCA, Spr.
9. Festuca Donax, Prodr. MS.

F’. panicule large, diffuse, subsecunde, nutantis ramis elongatis,
flexuosis: spiculis 3-floris, lineari-lanceolatis, glomeratis; flosculis glabris,
Vol. IV. Part I. B

10 Mr. Lowe on the New Plants and Land Mollusca

linearibus, muticis: foliis omnibus planis, elongatis, acuminatis; margi-
nibus scabris: culmo vaginisque glabris: ligula exserta, ovata, acuta:
radice fibrosa, perenni.

Hab. in Madere convallibus.

Gramen giganteum, 3—4 pedale, sylvaticum.

10. Festuca albida, Prodr. MS.

F’. densé cespitosa: panicule lanceolata, elongate, contracte, erec-
tiuscule rhachi ramisque pubescentibus: spiculis puberulis bifloris; flos-
culis calyce longioribus, muticis: foliis conduplicatis, elongatis, scaber-
rimis, serrulatis: culmo superné vaginisque pubescentibus; ligula ab-
breviata: vaginarum oris ciliatis: radice perenni.

Hab. in Maderz convallibus.

Sylvatica, bipedalis. Culmorum bases rudes, crassissimi, perennes,
glomerato-cespitosi. Folia culmos sub-equantia, numerosa. Panicula
pallida, albida. Spicule cum rudimento pedicellato flosculi tertii.

Habitus omnind Festuce. .

Orv. IV. CYPERACEE.
Gren. CAREX, Linn., Spr.

§§. Spicis plurimis; lateralibus androgynis, pedunculatis; terminali mascula.
11. Carex myosuroides, Prodr. MS.

C. spicis ¢ (apice masculis) sub-septenis, remotissimis, solitarlis, cy-
lindricis, utrinque attenuatis, densifloris, demtm pendulis, gracilibus,
elongatis, simplicibus, inferioribus pedunculatis; pedunculo vaginam
duplo excedente: stigmatibus tribus: fructibus, levibus, minimis, tri-
quetro-oblongis, squamas lanceolatas, acuminatas «quantibus; rostro
brevissimo, obtuso, integro, spits incurvo: culmo triquetro, levi.

Hab. in Madere ora septentrionali; ad margines rivulorum, sca-
turigines, &c,

of Madera and Porto Santo. rf

12. Carex elata, Prodr. MS.

C. spicis ¢ (apice ¢) sub-senis, remotissimis solitariis, linearibus,
laxifloris, pendulis, gracilibus, elongatis, basi compositis, ramosis, omni-
bus inclusé pedunculatis sc. vagina pedunculum sub-excedente: stigma-
tibus tribus: fructibus costatis, obovato-triquetris, rostratis, squamas
oblongo-ovatas, aristatas sub-equantibus; rostro tenui, recto, bifido,
levi: culmo triquetro, levi.

Hab. in Maderz convallibus umbrosis; sylvatica.

Orv. V. ASPARAGE.
Gren. ASPARAGUS, Linzn., Spr.
13. Asparagus scoparius, Prodr. MS.

A. caule frutescente, inermi, erecto, virgato ramisque patentibus, tere-
tibus, levibus: foliis fasciculatis, erecto-patentibus, teretibus, setaceis,
levibus, sub-mucronatis, sub-pungentibus: pedunculis densé fasciculatis,
foliis sub-brevioribus.

Hab. in rupibus Madere.

14. Asparagus scaber, Prodr. MS.

A. caule frutescente, inermi: ramis patentissimis, subdeflexis: foliis
fasciculatis, rigidis, pungentibus, patentissimis, spe deflexis, ramulisque
inequalitér angulatis, scabris: floribus fasciculatis; pedunculis foliorum
dimidium zquantibus.

Hab. in Madere rupibus.

Gren. RUSCUS, Linn., Spr.
15. Ruscus Hypophyllum, Linn.
a. latifolius ; foliis ovalibus, latioribus, 7—nerviis, distichis.
R. Hypophyllum, Bot. Mag. ¢t. 2049.

Hab. in Maderz convallibus umbrosis.
B2

12 Mr. Lowe on the New Plants and Land Mollusca

8. lanceolatus; foliis lanceolatis, angustioribus, numerosis, 5—nerviis;
inferioribus verticillatis; caule elatiore.
Hab. in convallibus umbrosis Madere.
Species forsan. Caules 2—3 pedales, superne foliosi. Folia 9—16,
inferiora 5—6_ pollices longa, 2—24 lata; superiora disticha. Verti-
cillus imus 3—6 folius. Cetera feré ut in a.

Orv. VI. SMILACE.
Gren. SMILAX, Linn., Spr.
16. Smilax pendulina, Prodr.. MS.
iS. caule fruticoso, scandente, sub-aculeato, tereti: aculeis caulinis
raris, sparsis, abbreviatis, deflexis: foliis inermibus, coriaceis, rigidis, un-
dulatis, 7—9 nerviis, venoso-reticulatis, laté cordatis, acuminatis; peti-
olis compressis, supra canaliculatis, basi 2-cirriferis: racemis flexuosis,
geniculatis, longissimis, filiformibus, pendulis, terminalibus, paniculatis.
ramosis; floribus ad genicula fasciculatis: baccis subglobosis, “rubris.”
Smilax latifolia, So/. ISS. et Herb. Banks! non R. Br.
Hab. in rupibus Madere.

Flores albi, racemis elegantissimis dispositi; feminei masculis paullo
majores. Baccas rité maturas non vidi; rubras incole ferunt.

Orv. VII. DIOSCOREE, R. Br.
Grn. TAMNUS, TYourn., Juss.
Tamus, Linn.

17. Tamnus edulis, Prodr. MS.

Lusitanicé, “ Norsa.”

Anglice, “ Porto Muniz Yam.”

T. foliis cordato-acuminatis, 9-nerviis: stipulis sub-nullis: racemis

elongatis: floribus sub-remotis; petalis ovalibus; stigmatibus simpli-
cibus.

of Madera and Porto Santo. 13

Dioscorea sativa, Bowdich, Hac. in Mad. p. 115.

Hab. in Madera.

Radix magna, extrinsécus pallidé brunnea, intts alba, flavescens :
sapore miti, edulis. Flores diceci, purpurascentes, luridi. Bacce dia-
metro ;poll. elliptice, “rubra.” In parochié “Porto Muniz” dicta,
Caurum versus, sola colitur: incole plantam indigenam credunt; ipse
nunquam nisi plané cultam aut ex cultu ortam vidi. In Canariis,
monente amico P. B. Webb, arm., procul dubio indigena; unde forsan
in Maderam introducta est.

Radix in aqua multas horas (X ad XII.) bulliente, tandem coctilis.

Orv. VIII. ORCHIDE.
Gen. ORCHIS, R. Br.

18. Orchis foliosa, Sol. MSS.

O. tuberibus palmatis: labello trilobo, subplano, expanso, latiore
quam longo; lobo medio lateralibus rotundatis, crenulatis angustiore,
obtuso, integro: sepalis obtusiusculis; exterioribus erectis; duobus in-
terioribus reflexis: germine cornu descendens, tenue, zquale, obtusum
superante: bracteis foliaceis, flores zquantibus: caule solido, elato.

Orchis foliosa, Sol. MSS, Masson, et Herb. Banks!

Hab. in umbrosis convallium sylvisque Madere.

Ab Orchide longibracteatd Bivon, (Bot. Reg. t. 357): quacum a
nonnullis confusa, omnind distincta. Flores magni, purpurei, inodori.
Caulis 2—pedalis.

Gren. GOODYERA, R. Br.
19. Goodyera macrophylla, Prodr. MS. Tab. I. ff. 1—12.

G. perianthii campanulati labello glabro cochleari-calceolato; se-
palis tribus exterioribus pubescentibus: columna anticé acuminata:

massis pollinis lineari-clavatis: spica pubescente: floribus secundis, brac-
teas superantibus: foliis ovalibus, reticulato-nervosis: caule repente.

14 Mr. Lowe on the New Plants and Land Mollusca

Hab. gregaria in declivibus sylvarum Madere humidis, umbrosis.
Rariss.

Caules repentes; demim erecti, pedales; superné cum bracteis,
germinibus, sepalisque tribus exterioribus pallidé ferrugineo-pubescentes.
Folia sat magna se. semipedalia, 3 poll. lata. Spica secundiflora, primim
pyramidata. Flores conferti, inodori, albidi, sub-cernui, ~ poll. longi.

Crass. III. DICOTYLEDONE.

Orv. I. AMENTACEE.
Gen. SALIX, Linn. Spr.
* Amenta precocia.

20. Salix canariensis, Sm.

§. arborescens, ramis glaucis, pruinosis, petiolisque tomentosis :
foliis lanceolatis, elongatis, utrinque attenuatis, sub-integerrimis; supra
glabriusculis, lucidis; subtts glauco-incanis, sub-tomentosis: stipulis mi-
nutis, adpressis, ovatis, crenatis: squamis ovato-oblongis, obtusiusculis,
sub-spathulatis, sericeo-villosis: germinibus magnis, pedicellatis, ovato-
lanceolatis, acuminatis, styloque abbreviato glabris: stigmate utroque
demum bifido.

Hab. in rupibus madidis Madere: etiam Nivarie, P. B. Webb,
arm.

Arbor feré 20 pedes alta evadit. Ramuli crassi, seepe colorati.
Gemme magne. Amenta ¢ cylindrica, abbreviata, 2-andra; ¢ elon-
gata, graciliora. Ex characteribus videtur S$, pomeranice Willd. affinis.

of Madera and Porto Santo. 15

GEN. QUERCUS, Linn.

21. Quercus mitis, Herb. Banks.

Q. foliis ovatis, subcordatis, obtusis, integriusculis, sinuolato-denti-
culatis, dentibus remotis, obsoletis; subtus, petiolis, ramulisque incano-
tomentosis.

Quercus mitis, Herb. Banks!

Hab. in “ Madera, Donne 1776:” Herb. Banks.

Ramuli (in specimine Banksiano) sub-umbellati; juniores tomento
brevissimo, cinereo obducti. Petioli sub-semipollicares. Folia 1j—2
poll. longa; 1;—1} lata; alterna, ovata, obtusissima, basi sub-cordata,
integriuscula s: marginibus sub-sinuolatis, nervisque lateralibus in denti-
culos remotos, obsoletos excurrentibus; coriacea, venosa, venis subtiis
prominentibus; supra lucida, glaberrima; subtis cum petiolis tomento
brevissimo densé velutina, in junioribus albo-incano, demim sub-ferru-
gineo. Flores masculi sessiles, glomerati, in spicis abbreviatis se: vix
uncialibus, axillaribus, inferioribus congesti: Feminei pauciores, spicati,
vel solitarii pedicellati, superiores sc: in axillis foliorum terminalium
versus apices ramulorum supra masculos nascentes. Rhachis spicarum
calycesque tomentosi vel lanuginosi.

Fructus in specimine deest.

Orv. II. URTICE.
Gen. URTICA, Linn. Spr.

22. Urtica elevata, Prodr. MS.

U. caule suffruticoso, lignoso, foliisque oppositis, longé petiolatis,
cordato-ovatis, grossé dentatis, lucidis, glabris: petiolis filiformibus
sparsim setosis: spicis axillaribus, pedunculatis, filiformibus, simplicibus,

interruptis, paucifloris, laxis, folio multtm brevioribus,
Urtica elevata, Herb. Banks !

Hab, in rupibus convallium Maderz.

16 Mr. Lowe on the New Plants and Land Mollusca

Rami tenues, diffusi, debiles. Folia sat magna, ad apices ramo-
rum sub-conferta. Habitus omnino generis. Planta inermis (haud urens
se. pungens).

Gen. PARIETARIA, Linn., Spr.
23. Parietaria gracilis, Prodr. MS.

P. lucida, pubescens: caulibus ramisque gracilibus, erectis: foliis
rhombeo-ovatis, obtusis, 3-nerviis, petiolatis; petiolis filiformibus, folia
equantibus: glomerulis axillaribus; floribus pedunculatis, sub-cymosis,
1—2—3 bracteatis; bracteis (seepitis 3) angustis, lanceolatis, calyce 4—fido
brevioribus, post anthesin glanduloso-pubescentibus, inzequalibus, 1—2
dilatatis, foliaceis, calycem superantibus.

Hab. in Madera; rariss.

Orn. III. LAURINEE.
Gren. LAURUS, Spr.

24. Laurus Barbusana, Prodr. MS. .

L. foliis perennantibus, lanceolato-oblongis, utrinque attenuatis,
coriaceis, rigidis; supra nitidissimis; infra axillis venarum nudis
(e-glandulosis): pedunculis ad ramulorum apices congestis, paniculatis,
sub-racemosis: pedicellis sub-elongatis, laxis: floribus hermaphroditis ;
calycibus sexfidis.

Hab. in Madere Sylvis.

Arbor magna. Folia sepe cymbiformia. Anthere biloculares.
Drupa ? pollicis longa, 4 lata; non calyculata.

Orv. IV.§ CHENOPODIE.
Gen. ATRIPLEX, Linn., Spr.
25. Atriplex parvifolia, Prodr. MS.

A. suffruticosa, procumbens, farinoso-incana: foliis confertis, alter-
nis, ellipticis vel oblongis, repandis, sub-sinuato-erosis vel integris: valvis
hastatis, integerrimis, dorso muriculato-tuberculatis.

of Madera and Porto Santo. 17

Atriplex portulacoides (angustifolia) Herb. Banks!

Atriplex portulacoides var. angustifolia, So/. MSS’!

Hab. in Insula Portis S". In Canariis, P. B. Webb, armig.

Species videtur: Cum A. portulacoide vera spe forsan confusa.
Cf. A. portulacoiden Desf: Fl. Ail. U1. p. 392. An hue quoque spectat
A. verrucifera B. angustifolia Bieberst?

Orv. V. NYCTAGINES®.
Gen. MIRABILIS. Linn, Spr.

26. Mirabilis divaricata, Prodr. MSS.
M. floribus congestis, terminalibus, sub-pedunculatis: corollé calycem
sextuplo superante; tubo longissimo, pubescente; limbo plicato (laciniis
emarginatis) tubi quartam partem aquante: foliis sub-cordatis, petiolatis ;
supra, petiolis, lineaque caulina utrinque exarata sub-pubescentibus : ramis
dichotomis, nodosis, cauleque erectis: pericarpio rugoso, glabro, (atro).

Mirabilis hybrida, Lepel/? sed folia in planta Maderensi (quamvis
lucida) minimé glabra, &e.

Hab. in hortis et ruderatis Madera. Circa urbem Funchalensem
nune quasi indigena.

Valde ramosa, 3—5 pedalis; ramis divaricatis, demum corymbosis
vel convexo-fastigiatis. In M. Jalapé vera caules multo humiliores,
minus ramosi; pericarpia minora, ferrugineo-pubescentia, minus rugosa,

egranulata.

Orv. VI. PLANTAGINES.
Gren. PLANTAGO, Linzn., Spr.
27. Plantago leiopetala, Prodr. MS.

P. caulescens: caule abbreviato, basi frutescente: foliis confertis,
lanceolatis, utrinque attenuatis, nervosis, glabriusculis, nitidis, integerri-
mis: pedunculis folia superantibus, angulatis, glabris: spicis abbreviatis,

Vol. IV. Part I. (o

18 Mr. Lowe on the New Plants and Land Mollusca

oblongis ovatisve, obtusis, nudis: laciniis calycinis latis, scariosis, cari-
natis, corollisque glabris.
Hab. in cacuminibus Ins®: Portis 8".

Plantagini lanceolate proxima; cultura non mutatur.

Orv. VII. PLUMBAGINE.
Gen. STATICE, Spr.
28. Statice pyramidata, Prodr, MS.

S. cespitosa, glauca: scapo erecto, ramoso, aphyllo: foliis radica-
libus, parvis, obovato-oblongis, acutis, mucronulatis, in petiolum atte-
nuatis, enerviis: panicule pyramidate ramis patentissimis, recurvis:
floribus conglomerato-imbricatis; laciniis ealycinis obtusiusculis.

Hab. in rupibus maritimis Ins*: Portis 5°.

Flores pallidé cerulei, parvi, glomerulis congesti. S. awriculefolie
et oleefolie affinis. A S. spathulatéd Desf. differt foliis acutis; scapo
magis ramoso; ramulis gracilioribus, sub-deflexis; floribus glomerulatis,
minoribus, &e.

Orv. VIII. LABIATA.
Gren. SALVIA, Linn., Spr.
29. Salvia collina, Prodr. MS.

S. caule herbaceo, viscoso-piloso: calyce 5-dentato, S foliis pin-
natifidis, incisis, vel sub-sinuatis, dentatis, venosis, glabris, leviusculis :
bracteis sub-rotundis, latis, cordatis, abbreviatis, acutis, inconspicuis, ca-
lycis dimidium xquantibus, integerrimis: verticillis 6-floris: corollis
calycem dupld superantibus: galeé falcata, compressa : lobo medio labii
inferioris cucullato; lobis lateralibus reflexis, parvis, abbreviatis, rotun-
datis, obtusis.

Salvia verbenacoides, Brot? (polymorpha, Hoffm.)

Hab. in collibus pascuisque altis Mader.

of Madera and Porto Santo. 19

Salvie pratensi et Verbenace (potissimum priori) affinitas summa ;
sat verd distincta. A Salvia bicolori Desf: differt caule humiliore; fo-
liis glabris; bracteis abbreviatis, latis. acutis; lobis lateralibus _labii
inf*. rotundatis, &c.

Gren. THYMUS, Linn.
30. Thymus micans, Sol. MSS.

T. pedunculis ad apices ramulorum congestis, sub-racemosis, axil-
laribus, oppositis, solitariis, unifloris: calycis $ labio superiore lato, ob-
soleté tridentato, marginibus recurvis; inferiore dentibus duobus sub-
erectis, lanceolatis, acutis, contiguis, equalibus profundeé inciso: bracteis
linearibus foliisque lineari-spathulatis, obtusis, basi attenuatis, pilisque
raris, longis, patentissimis, remotis, pectinato-ciliatis: caulibus hispidis,
prostratis, cespitosis, basi fruticulosis.

Thymus micans, Herb. Banks. et Sol. MSS'!

Hab. copiosissimé, cespitem efficiens, per totum campum illum
excelsum (5000 ad 6000 pedes altum) “Paul da Serra” ‘dictum, Madere.

Grn. SATUREJA, Linn.
31. Satureja thymoides, So/. MSS.

S. pedunculis axillaribus, multifloris, umbellatis: bracteis setaceis,
fasciculatis: dentibus calycinis dimidium tub sub-zquantibus: foliis
oblongo-lanceolatis, acutis, utrinque attenuatis, margine revolutis, sub-
puberulis, subttis sub-incanis: ramulis junioribus sub-pubescentibus; caule
fruticuloso, erecto.

Satureja thymoides, Sol. MSS. et Herb. Banks!

An Thymus therebinthaceus, Willd. Enum. Pl. Hort. Berol.
p. 6242

Hab. in Madera et Portu S$"; vulgaris.

Fruticulus elegans. Folia sub-micantia, odorata. [lores sub-con-
spicui, purpurascenti-rosei. Stamina tubo corolla breviora, inclusa,

20 Mr. Lowe on the New Plants and Land Mollusca

Calyces striati, sub-pubescentes; fauce villis clausa; dentibus setaceis.
sub-zqualibus, duobus inf*. aliquando sub-longioribus. Folia latitudine.
&e. variabilia; ideoque 7’ therebinthaceus Willd. (si ex descriptione ju-
dicare licet), vix nisi genere discrepans, idem videtur.

Orv. IX. PERSONATAE.
Gen. EUPHRASIA, Linn., Spr.

32. Euphrasia Holliana, Prodr. MS.

E. \aciniis calycinis, foliisque lanceolato-oblongis, obtusis; inferio-
ribus grossé dentatis; summis sub-integris: corolla (lutea) calycem duplo
excedente; staminibus corollam sub-zquantibus: caule ramoso.

Bartsia viscosa, var. foliis linearibus obtusis &c. Herb. Banks !

Hab. in sylvis Madere.

Corolla conspicuz, sat magne; labium inferius $-lobum, lobis ro-
tundatis, obtusissimis, denticulatis; superius simplex. Folia et habitus
quodammodd Euphrasie Odontitis.

Ob antheras distinctissimé aristatas, vera Huphrasie species: quin-
etiam toto habitu, caule ramoso &c. 4 Bartsid viscosd differt. Ab
Euphrasid luted, cui affinitate propior, dignoscitur caule ramisque ro-
bustis, nec filiformibus; foliis multd latioribus majoribusque, inferiori-
bus grossé vel inciso-dentatis: floribus multd majoribus, &c.

Nomen dedi in honorem amici F'r. Holl, botanosophi Germanici,
indefessi plantarum Maderensium indagatoris.

Gren. SCROFULARIA, Linzn., Spr.
33. Scrofularia racemosa, Prodr. MS.

S. foliis sub-cordato-oblongis, acutis, sub-duplicato-serratis, utrinque
cauleque acutangulo glabris, basi inequalibus; inferioribus appendicu-

of Madera and Porto Santo. 21

latis: thyrsi elongati aphylli ramis racemosis; racemis elongatis, flex-
uosis, patentibus, laxis, ramis pedicellisque sparsim sub-glandulosis: ca-
lycis glabriusculi laciniis obtusis: corolle labii inferioris lobo intermedio
revoluto, vix prominulo, minuto; superioris, rudimento staminis 5°.

squameformi, plano, rotundato.
Serofularia auriculata. Linn., Spr. &c. 2

a. longifolia; glaberrima, foliis acuminatis, elongatis, simplicitér serrato-

crenatis.

B. puberula; foliis radicalibus et junioribus subtis petiolisque pube-
rulis.

Hab. a. et. #. ad rivulos et in rupibus madidis convallium Ma-
dere.

Scrof. sulphurea Mill. Dict.; et S. Balbisii Wild. (ex Spr.), saltem
ex descriptionibus, equé plant nostre a. pertinere possent. S. awricu-
lata vera (cui prior forsan synonyma), foliis “ obtusis subtus hirsutis”
Spr. (“tomentosis” Linn.) “lobo terminali cordato aut ovato” Desf:
differre videtur. S. auriculatam Brot. Fl. Lusit. I. p. 201 vero, ob fo-
lia “subtus glabra,” ab a. nostra alienam egré putarem, nisi quod a
nonnullis ad S. trifoliatam Linn. relatam video. In re tam dubia, plan-
tam Maderensem pro tempore distinctam servandam putavi.

34. Scrofularia hirta, Prodr. MS.

S. foliis cordato-oblongis, acutiusculis, basi sub-aqualibus, excisis,
arguté duplicato-inciso-serratis, utrinque petiolis cauleque acutangulo
villosis; petiolis latis, sub-alatis, ex-appendiculatis, hirtis; thyrsi aphylli
ramis trichotomo-racemosis; racemis elongatis, pedunculisque glanduloso-
pubescentibus; calycis glaberrimi laciniis obtusissimis; corolla labii in-
ferioris lobo intermedio revoluto, vix prominulo, minuto; superioris,
rudimento staminis 5". papilleformi, minutissimo, brevi; genitalibus
exsertis.

Hab. in Maderz umbrosis humidis obscuris. Rariss.

22 Mr. Lowe on the New Plants and Land Mollusca

Orv. X. CONVOLVULE.
Gen. CONVOLVULUS?

35. Convolvulus? solanifolius, Prodr. MS.

C? caule volubili, fruticoso: foliis cordatis, ovato-oblongis, acutis,
integerrimis, petiolatis; junioribus, ramulis, petiolisque pubescentibus :
pedunculis axillaribus, solitariis, petiolo longioribus, apice sub-trifloris,
pedicellisque elongatis, nudis: calycibus ovalibus, obtusiusculis; .... .

Hab. in Madere rupibus. Rariss.

Corollam nondum vidi.

Orv. XI. SAPOTE.
Gey. SIDEROXYLON, Spr.

36. Sideroxylon Mermulana, Prodr. MS. “ Mermulana,’ incolarum.

S. inerme: foliis obovatis, obtusis, spathulatis, integerrimis, cori-
aceis, nervosis, lucidis, utrinque glaberrimis : pedunculis unifloris, ad
axillas aggregatis, brevibus, calycibusque velutinis.

Sideroxylon Mermulano, Herb. Banks. !

Hab. in rupibus, presertim maritimis, Madere.

Frutex, vel sub-arboreum. Flores parvi, pallidé carnei. Fructus
ruber, edulis.

Nomen “ Mermulana” in Canariis plante distinctissime, sed quoad
habitum simillime, se. Myrsinet canariensi, impositum scribit amicus
P. B. Webb, arm.

Orv. XII. COMPOSIT &.
* CICHORACE.
Gen. SONCHUS, Linn., Spr.
37. Sonchus ustulatus, Prodr. MS.

S. glaberrimus: caulibus simplicibus, brevissimis, herbaceis, basi
sub-lignosis: foliis radiatim confertis, decursivé runcinato-pinnatis, sub-

of Madera and Porto Santo. 23

carnosis, rigidis, subtus prasertim inter venas pulchré glaucescentibus ;
foliolis acutis, angulatis, sub-integerrimis vel dentibus sparsis, raris, mi-
nutis, callosis: scapi terminalis, aphylli, pauciflori, ramis raris, divari-
catis, solidis, pedunculisque 1-floris, supra incrassatis, paniculatis, nudis ;
squamis anthodii purpureo-nigricantibus, adpressissimis, latis.

S. hyoserifolius, (Hornem:) Spr?

a. angustifolia; foliolis angustis, confertis, acuminatis, margine poste-
riore sub-integerrimo.

Sonchus dentatus, Herb. Banks?

B. latifolia: foliolis majoribus, latioribus, distantibus, angulatis, utrinque
denticulatis: foliis supra vix glaucescentibus, profundits incisis.

“ Sonchus squarrosus ” (lined atramenti per medium verborum squar-
rosus (3 ducta, et “fruticosus” supra scripto) “et MSS. differt pani-
cula dichotomaé—planta minor. Madera Fr. Masson.” Herb. Banks!

Hab. in rupibus maritimis aridis Madere.

B (vix var.) est status potitis € solo vel humidiore vel magis um-
broso ortus.

De Soncho hyoseridifolio hue rite referendo, suspensus hereo.
Characteres feré iidem; nisi qudd illa inter “ fruticosos” wmumeratur,
cum nostra planta certissimé “ herbaceis” releganda est. An descriptio
cl. Sprengelii & specimine manco, (se. sine radice desiccato), in her-
bario servato, (quale est forsan Sonchus dentatus Herb. Banks.), facti-
tata; ideoque erronea? Nam in tali, caulis casu forsan quodam ligno-
sus evadere posset: specimine ttim speciem ramuli fruticis cujusdam
ramos omnino prebente.

S. dentatus Herb. Banks. in omnibus nisi caule lignoso cum vari-
etate a. nostra convenire videtur. Sed, ciim preter hoc, alia exstant
specimina in Herb. Banks., ad (3. nostram certissimé pertinentia, que
a cl. Solandro ad alteram (quamvis revera distinctissimam et longé
alienam) speciem (S. squarrosum) referuntur, ideoque 4 S. suo dentato
plane distinguuntur, impensits suspicandum est hune S. dentatum a

24 Mr. Lowe on the New Plants and Land Mollusca

nostra planta alienum esse, et forsan S. hyoseridifolii (Hornem.) Spr.
veri synonymam. In re tam incerta, difficultatem minus nomine plané

novo quam veteri incerto augeri putavi.

Gren. TOLPIS, Gaert.
38. Tolpis crinita, Prodr. MS. Tab. Il. ff. 1—3.

T. caule ramoso: ramis virgatis: foliis radicalibus humifusis, solo

adpressis, plerumque sinuato-pinnatifidis, sub-canescentibus ; caulinis an-
gustis, lanceolato-linearibus: bracteis setaceis, abbreviatis, ad apices pe-
dunculorum inferné nudorum congestis: seminibus (omnibus, radii sc.
conformibus) sub-quadrisetis.

Crepis crinita, Sol. MSS. et Herb. Banks!

Crepis incrassata, Herb. Banks. “Insula Azores F ayal Mess*™,
Forster” !

Hab. in Madere collibus apricis. —

Gren. CREPIS, Spr.
39. Crepis pectinata, Prodr. MS.

C. caule frutescente, ramoso, foliato; ramis diffusis, virgatis: fo-
liis flaccidis, tenuissimé et profundé divisis, pectinato-pinnatifidis; laci-
niis distantibus, elongatis, lineari-filiformibus, supra glabris, subtus sub-
farinosis: pedunculis proliferis, superné incrassatis, squamosis, squamisque
minutis, erectis, anthodioque farinoso-albescentibus.

Crepis tenuifolia, Sol. MSS. et. Herb. Banks! non Willd.

Hieracium fruticosum foliis tenuissimé coronopi modo divisis.
Sloan. Cat. 123.— Hist. Jam. p. 19. t. 5. f.1, 2. (Icon. mala; Deser. opt.)

Hab. in rupibus apricis Mader ubique.

Affinitate et habitu Crepidi succulente Hort. Kew: (C. coronopi-
folie Desf:) proxima; cujus speciei, pre ceteris polymorphe, (in Ma-
dera vix mints vulgaris), varietatem esse meram alia forsan dies do-
cebit. In illé tamen, quamvis folia aliquando profundits pinnatifida

of Madera and Porto Santo. 25

quam in statu normali, nunquam ut in C. pectinatd nostra feré fili-
formia et tenuitér divisa, laciniis elongatis linearibus (feré ut in Core-
opside tinctorid Hort.) videntur.

Oss. C. succulenta Ait. et C. pectinata nob: genus Tolpidem cum
Crepide arctissimé conjungunt.

40. Crepis macrorrhiza, Prodr. MS.

C. glaberrima: radice perenni, crassa, carnosa: caulibus solidis, fo-
liatis, simplicibus, superné paniculatis: foliis omnibus indivisis, oblon-
gis, dentatis, sessilibus, nitidis, sub-carnoso-coriaceis: panicula larg4, mul-
tiflora; pedunculis superné sub-incrassatis, squamosis; anthodiis sub-
farinoso-pubescentibus.

Crepis macrorrhiza, Herb. Banks: et Sol. MSS! Hook. in Bot.
Mag. t. 2988!

Hab. in Maderz rupibus.

41, Crepis? andryaloides, Prodr. MS.

C? glanduloso-hispida: radice carnosa, bienni: caule sub-fistuloso,
foliato, simplici, superné laxé paniculato, hispido: foliis omnibus in-
divisis, oblongis, acuminatis, undulatis, remote runcinato-dentatis, sub-
sinuatis, sessilibus, hispidis: floribus laxé paniculatis, remotis: pedun-
culis nudis, gracilibus, divaricatis, laxis, anthodiisque cylindricis, glan-
duloso-hirsutissimis: involucro erecto, persistente.

Hab. in convallibus Madere.

Semina matura non vidi; pappus in immaturo revera sessilis: sed
cum in veris quibusdam Borkhausie speciebus pappus in semine im-
maturo omnino sessilem vel subsessilem videre licet, in nostra forsan
planta pappus seminis maturi stipitatus evadit.

Gen. BORKHAUSIA, Bohm. Spr.
42. Borkhausia laciniata, Prodr. MS.

B. radice annua: caule erecto, stricto, ramoso, paniculato, sub-
puberulo, nitido: foliis laciniato-pinnatifidis, vel runcinato-dentatis, sinu-
Vol. lV. Part I. D

26 Mr. Lowe on the New Plants and Land Mollusca

atis, glabris; radicalibus plerumque integriusculis, oblongis; caulinis
lineari-lanceolatis, semi-amplexicaulibus, basi auriculatis, sub-sagittatis,
dentato-laciniatis: floribus corymboso-paniculatis: anthodii squamis dorso
inferné nigrescenti-glanduloso-hispidis, interstitiis sub-farinoso-puberulis :
squamis involucri laxis, farinoso-puberulis.

a. pinnatifida; foliis profundius divisis.
Crepis biennis, Herb. Banks! quoad specimina in Madera a Masson
lecta.
“Crepis Dioscoridis” (linea per verbum Dvoscoridis ducta) “ L. var.
corolla undique lutea. Madeira Fr. Masson 1777.” Herb. Banks!
B. integrifolia; foliis integriusculis.
“Crepis Dioscoridis” (lined per verbum Dioscoridis ducta) “ L. var.
foliis margine nudis, Madeira Fr. Masson 1777.” Herb. Banks!

Hab. in Madera; in vinetis, locis cultis, frequens.

43. Borkhausia divaricata, Prodr. MS.

B. yradice crassa, fusiformi, bienni (perenni?): caulibus ramosis,
paniculatis, solidis, inferné glabris, superné pedunculisque divaricatis,
patentibus, hispido-glandulosis: foliis rigidis, glaberrimis, undulatis;
radicalibus sinuato-runcinatis, caulinis basi semi-amplexicaulibus, dila-
tatis, ovato-acuminatis, integriusculis: floribus sparsis, paniculatis: an-
thodiis post anthesin ovatis, basi ventricosis; squamis basi hispido-
glandulosis, superné squamisque laxiusculis involucri glabris.

a. robusta; caulibus erectis, virgatis, pedalibus, miultifloris, foliosis:
foliis seape runcinato-pinnatifidis.

Hab. in Promontorio S“. Laurentii Madere.

6. pumila; caulibus sepe diffusis, glabriusculis, paucifloris, plerumque
nudis: foliis radicalibus indivisis, integriusculis vel runcinato-sinu-
ato-dentatis, sub-carnosis: anthodiis hispidioribus.

Hab. in Portii S"°.—Status potits, ex solo aridiore, quam varietas
praecedentis.

of Madera and Porto Santo. 27

44. Borkhausia hieracioides, Prodr. MS.

B. yadice annua?: caule erecto, ramoso, paniculato, foliarum costa
centrali, pedunculis, anthodiisque setoso-hispidis vel sub-muricato-spinel-
losis: foliis glabris, indivisis, denticulatis, denticulis raris, sparsis, mi-
nutis, subulatis; radicalibus lanceolato-oblongis, acutis, basi attenuatis ;
caulinis ovato-acuminatis, basi dilatato-auriculatis, semi-amplexicaulibus :
floribus corymbosis; flosculorum ligulis elongatis, laxis, patentissimis,
sub-pendulis.

a. integrifolia; foliis integriusculis, sub-sinuolatis.

B. pinnatifida; foliis sub-pinnato-runcinatis.
Hab. in Madere ora Septentrionali.

45. Borkhausia dubia, Prodr. MS.

B. radice bienni: caule erecto, stricto, @ basi ramoso, ramisque
foliatis, costaque centrali foliarum subtts hispidis: foliis lucidis, gla-
bris, indivisis, marginibus undulatis, sinuato-runcinatis et denticulatis,
denticulis intermediis plurimis, inequalibus, subulatis vel ciliato-seta-
ceis; radicalibus elongatis, oblongo-lanceolatis, acutis, basi attenuatis;
caulinis basi cordato-zqualibus, amplexicaulibus, oblongis, acuminatis ;
summis linearibus, sub-integerrimis, setaceo-ciliatis: floribus sub-corym-
bosis: ramulis superné, pedunculis, anthodiisque dense glanduloso-pu-
bescentibus, sub-incanis, farinaceo-puberulis: pappo sub-stipitato.

Hab. in convallibus Madere.

Precedenti proxima; differt autem habitu distinctissimo, ramis
superné, pedunculis, anthodiisque densé glanduloso-pubescentibus, sub-
incanis, farinaceo-tomentosis; floribus minoribus in corymbos laterales
collectis, floseulorum ligula nee elongata nec pendula, pappo sub-stipi-
tato (quo ad Crepides veras accedit), foliarum margine inequaliter sed
conspicué et in omni parte runcinato-sinuatis, caulinis basi aqualibus,
cordatis, (non dilatato-auriculatis).

46. Borkhausia comata, Prodr. MS.

B. radice fusiformi, carnoso: caule erecto, é basi ramoso, foliato,

hirto-setoso: foliis indivisis, denticulatis; radicalibus glabris; caulinis
D2

28 Mr. Lowe on the New Plants and Land Mollusca

summis ciliato-crinitis: floribus corymbosis: anthodiis hirsutissimis, co-
matis; squamis crinitis.

Crepis comata, Herb. Banks. et Sol. MSS!

“ Hab. in Madere sylvis; Fr. Masson 1777.” Sol.

Pappus distinctissimé stipitatus.

Gen. THRINCIA, Roth, Spr.

47. Thrincia nudicaulis, Prodr. MS.

T. foliis hispidis, sub-dentato-sinuatis: pappo disci stipitato.

Leontodon nudicaule, Herb. Banks!

Hab. in apricis Madere ubique; vulgatissima.

Flosculorum tubulus ad apicem pilosus; ligulorum lacinize eglandu-
lose. Pappus radii paleaceus; disci plumosus. Semina sursim atte-
nuata, acuminata, in rostrum gracile, elongatum producta; unde pappus
stipitatus.

Thrincia hirta Hook. Brit. Fl. (Apargia hirta Sm. Eng. Fi., He-
dypnois hirta Ejusd. in Engl. Bot.) pappo disci sessili, potissimim
differt. Eodem charactere, necnon genere, sc. pappo radii paleaceo, ab
Apargid hispida omnind distincta.

* * CINAROCEPHALA
Gren. CIRSIUM, Tourn, Spr., &e.
(CNICUS, Aliorwm).
48. Cirsium latifolium, Prodr. MS.

C. inerme: foliis sessilibus, basi auriculatis, amplexicaulibus, om-
nibus elliptico-oblongis, latis, obtusis, indivisis, laté sinuato-crenatis,
setoso-spinelloso-ciliatis, supra lucidis, nudis, subtus cauleque lanato-
tomentosis, floccosis: pedunculis longissimis, floccosis, unifloris: antho-
diis sub-lanatis: squamis lanato-ciliatis, mucronatis, adpressis, inferiori-
bus ovatis, acutis; superioribus oblongis, obtusiusculis.

Carduus latifolius, Herb. Banks !

of Madera and Porto Santo. 29

Hab. in Madere convallibus.

Species pulchra, distinctissima, C. heterophyllo affinis.
Caulis 2—3-—pedalis. Folia ampla, subtis sepe nivea. Flores
purpurel.
** * CORYMBIFER®.
Gren. GNAPHALIUM, Linz., Spr. &e

49. Gnaphalium melanophthalmum, Prodr. MS.

G. fruticosum: foliis sparsis, sessilibus, lanceolatis, acuminatis, basi
attenuatis, ramisque niveo-tomentosis, canescentibus: paniculis termi-
nalibus, congestis, corymbosis: squamis anthodii nivei, globosi, laxis,
ovatis; inferioribus obtusis, rotundatis; superioribus acutiusculis.

Gnaphalium rupestre, Herb. Banks !
Oss. Gnaph. rupestre, Rafin: jam adest. Steud. Nom. Bot.
Hab. in rupibus convallium Madere.

Flores nivei, odori; disco post anthesin nigro.

Orv. XIII. RUBIACE®.
Gen. GALIUM, Linn, Spr.

50. Galium productum, Prodr. MS.

G. glabrum: foliis octonis, lanceolato-linearibus, acutis, cuspidatis,
reflexis, sub-integerrimis, denticulis marginalibus raris, obsoletis, antror-
sim spectantibus, utrinque levibus, supra cauleque pariim ramoso
lucidis: panicularum lateralium terminaliumque ramis divaricatis, ab-
breviatis: corolla laciniis obtusiusculis, mucronatis: fructibus levibus,
glabris: caule 4—angulari, debili, diffuso, elongato, simpliciusculo, levi,
basi suffruticoso.

Hab. in Madere saxosis, sepibus, rupibus &c. frequens.

30 Mr. Lowe on the New Plants and Land Mollusca

Orv. XIV. UMBELLIFERZ.
Gen. @NANTHE, Spr.
51. Qnanthe pteridifolia, Prodr. MS.

iS. radicibus tuberosis, fusiformibus, fasciculatis: caule erecto, in-
ferné tereti, levi, ramis angulatis, striatis: foliis omnibus tripinnatis ;
pinnis pinnulisque omnibus remotis, oppositis, patentissimis, distichis ;
foliolis ultimis ovatis lanceolatisque, acutis, inciso-dentatis pinnatifidis-
que, basi cuneatis: umbellis oppositifoliis; radiis inaequalibus; bracteis
paucis, subnullis, bracteolisque linearibus: fructu suberoso.

CE. apiifolia, Brot?

Hab. in rupibus madidissimis convallium Madere.

Radices repentes; tuberibus fusiformibus, fasciculato-filipendulis,
erassitie digiti. Caules elati, fistulosi, esculenti. Folia maxima, ele-
gantia, latevirentia, foliolis exiguis, tenuibus, concinnis. Unmbelle me-
diocres, sat parve; floribus albidis, aspectu eorum -dinanthes crocate.
Calyx persistens. Petala mucrone elongato, inflexo. “ Floral Recep-
tacle,” Sm. nullum. Stylopodia (* Bases of Styles,” Sm.) tumida, glo-
bosa. S*yli persistentes, post anthesin elongati, fructum maturum
equantes. Fructus ovato-oblongus, lateralitér (sc. sutura) compressus,
presuberosus. Mericarpia*, striis 7 dorsalibus, levibus, sub-zqualibus,
tribus vix majoribus; interstitiis angustis, planis, equis; sutura utrin-
que spatio tumidulo, latiusculo, levi, spongioso vel suberoso. Al-
bumen sive Perispermium teretiusculum, intus (plano-convexum), Vitte 6,
rect, wquales; 4 dorsales, quidistantes; jugis tribus dorsalibus
sub-majoribus alternantes, sc. 4 intermediis opposite; relique due

* Hee forsan melius ita describenda: mericarpia jugis 5, tribus dorsalibus filifor-
mibus, sub-prominulis; duobus lateralibus marginantibus dilatatis, spongiosis, spatium
latum, convexum, tumidulum, suberosum utrinque formantibus ; valleculis 1-vittatis, 1-striatis ;
striis filiformibus, distinctis, juga subzquantibus, sc. vix minis prominulis ; hine mericarpia
7-striata apparent. Carpophorum evanidum, sub-nullum.

of Madera and Porto Santo. 31

juncturam respicientes, approximate. Totius plant succus aquosus.
An Genus?

Gren. SAMBUCUS, Linn., Spr.
52. Sambucus nigra, Linn. Sm., &e.

a. communis; foliolis ovatis.
Sambucus nigra, Auct.
Hab. in Europa; Anglia, &c.
B. lanceolata: foliolis lanceolatis vel ellipticis, elongatis.
Sambucus lanceolata, Herb. Banks.
Hab. in Madere sylvis: in hortis etiam ab incolis colitur.

Preter foliola magis elongata, omnia ut in a; ideoque vix spe-
cies consenda.

Orv. XV. CRASSULACEZ.
Gen. SEDUM, D.C.

53. Sedum fusiforme, Prodr. MS.—Tab. 3. ff. 1, 2.

S. caule fruticuloso, ramoso; ramulis confertis, erectis, tortuosis,
glabris, inferné nudis: foliis omnibus sparsis laxis sub-patentibus, car-
nosis, crassis, fusiformibus, sub-teretibus, supra planiusculis, acutiusculis,
utrinque attenuatis, glaberrimis, glaucis: cymis terminalibus, cormboso-
fastigiatis, paucifloris: petalis 5, lanceolatis, obtusiusculis, patulis: squa-
mis nectariferis brevibus, lunatis.

Hab. in Mader rupibus excelsis aridis maritimis.

Ramosissima, cespitosa, humilis. Flores flavi.

S. altissimo proxima; habitu prorsis S. nudi, cui maxime affinis.
54. Sedum farinosum, Prodr. MS.

S. candicans: caulibus herbaceis, prostratis (repentibus?), elon-
gatis, inferné nudis, sub-simplicibus; foliis ad apices confertis, 4—fariis,

32 Mr. Lowe on the New Plants and Land Mollusca

caulibusque albo-farinosis, teretibus, supra sub-planulatis, obtusissimis :
cymis terminalibus, 3-partitis: petalis 5, ovato-lanceolatis, acutis; squa-
mis nectariferis

eee POC my eR APO) OS tach Oy Goth fy Chee UCR. he Grouch ete cc. Sify Pyne

carpellis rostratis, acutis.

Hab. in rupibus umbrosis Madera, ad altitudinem 4000 ad 5500
feré pedum.

Petala alba, nervo extra rubro.

Orv. XVI. LYTHRARIEZ.
Gren. LYTHRUM, D.C.
55. Lythrum junceum, Sol. ISS.

L. floribus axillaribus, hexapetalis, dodecandris: filamentis 6 brevis-
simis; 6 longioribus, tubo brevioribus; antheris sub-inclusis: calycis
angulati dentibus alternis minoribus: foliis alternis, confertis, lanceolato-
linearibus, sub-glaucescentibus: caulibus acuté 4~angularibus, debilibus,
humifusis, elongatis; deorstim nudis, suffrutescentibus.

Lythrum junceum, Sol. MSS! et Herb. Banks. quoad specimina
Maderensia !

acutangulum, Lagasca, (LL. Grefferi var? D.C.) ?
Hab. in Mader humidis, frequens.
Caules graciles, demiim prelongi, simpliciusculi, inferné nudi,

frutescentes. Folia parva, sub-conferta. Flores hexapetali, magnitudine
mediocri, leté purpurei.

Orv. XVII ROSACESX.
Gen. RUBUS, Linn. D.C.
56. Rubus grandifolius, Prodr. MS.

R. caulibus fruticosis, angulatis, aculeatis, glabris, procumbentibus,
sterilibus elongatis; aculeis sparsis, compressis, recurvis, numerosis: fo-
liis quinatis (rard ternatis), sub-pedatis; foliolis ovato-oblongis, acumi-

of Madera and Porto Santo. 33

natis, grossé duplicato-serratis, utrinque glaberrimis, nudis, longe peti-
olulatis; petiolis petiolulisque sparsim aculeatis: panicule elongate, ter-
minalis, ramis pedunculis calycibusque densé purpureo-glandulosis :
laciniis calycinis reflexis, inermibus, petalis multd brevioribus.

Rubus pedatus Herb. Banks! et Sol. MSS! non Smith.

Hab. in rupibus Madere.

Folia lucida, utrinque viridia, magna. Flores albi, conspicul,

magni. Fructus sat magni, atri.

Orb. XVIII. LEGUMINOS®.
Gen. VICIA, Tourn. D: C.

57. Vicia albicans, Prodr. MS.

V. annua, villosa, sub-canescens: caulibus tetragonis: cirris valde
ramosis: foliolis oblongis, mucronatis, numerosis, oppositis et alternis:
stipulis semi-sagittatis, inciso-dentatis: pedunculis sub-bifloris, folio mul-
tum brevioribus; floribus secundis, laxis, sub-remotis: dentibus calycinis
duobus superioribus minimis, obsoletis; inferioribus ovato-subulatis,
medio longiore; omnibus tubo brevioribus et cum toto calyce colo-
rato pilosis: stylis capitatis, infra capitulum globosum undique, subtis
vero presertim, barbatis: leguminibus oblongis, latiusculis, brevibus,
sub-compressis, albido-hirsutissimis, pendulis, sub-tetraspermis; semini-
bus globosis, viridi-fuscis, nigro-maculatis, glabris.

Hab. in rupestribus aridis apricis Madere.

Flores magnitudine mediocri, paulld sc. majores quam in V’. Craced,
rosei vel purpurei, apice purpureo-nigro, vexillo striato. V. atropur-
puree, Desf: ; trichocalyci, Moris.; Broteriane, Ser. in D.C. Prodr.
(V. villosa Brot non Roth.) affinis, Radice annua a V. perennt D. C.;
argented, Lapeyr.; variegatd Willd. ; alpestri Stev.; cinered Bieb., necnon
aliis notis distineta.

58. Vicia micrantha, Prodr. MS.

V. annua, gracilis, glabriuscula: caulibus filiformibus: cirris ra-
mosis: foliolis angusto-lanceolatis vel lineari-oblongis, remotiusculis,

Vol. IV. Part I. E

34. Mr. Lowe on the New Plants and Land Mollusca

ob-tusiusculis, sub-puberulis: stipulis parvis, angustissimis, semi-sagittatis,
superioribus simplicibus : pedunculis sub-bifloris, folio multtim brevio-
ribus: calyce leguminibusque latis, oblongis, compressis, 3—6-spermis,
villosis.

Vicia gracilis, Sol. MSS. et Herb. Banks! non Lovsl.

Hab. in Madera; Sol. e¢ Mass.

Foliola sub-octoparia. Flores perparvi, purpurascentes.

Gen. ONONIS, Linn., D. C.

59. Ononis dentata, Sol. IZS'S.—Tab. 4.

O. herbacea, annua, erecta, pilosa: foliis (omnibus) trifoliolatis ; fo-
liolis obovatis, serratis: stipulis ovatis, dentatis: floribus sparsis, soli-
tariis, axillaribus, pedunculatis, folio longioribus, cernuis: pedunculis
muticis: corolla calycem superante: laciniis calycinis 4 supremis antice
dilatatis, foliaceis (3—-) dentatis; infima simplici lineari-acuminata, in-
tegerrima; leguminibus calyce longioribus.

Ononis dentata, Sol. MSS! et Herb. Banks. quoad specimina 3 in

Insulis Canariis A. D. 1778 a cl. Masson lecta!

Hab. in Portu 8S”. “Insule Canariz Fr. Masson 1778,” Herb.
Banks: Yn apricis Nivarie, P.B. Webb, arm.

Flores conspicui; in plantis ab amico Rev.’ M. J. Berkeley in
Anglia cultis (4 seminibus que in Insulé Portis S". mense Maii, A. D
1828, ipse legi) vexillo roseo-purpureo, alis et carina pallidioribus; in
aliis (desiccatis) ab amico P. B. Webb arm. in Nivaria lectis, pallidé
flavi, carina purpurea.

Gren. ASTRAGALUS, D.C.
Series I]. OCHROLEUCI,
§. 7. . Bucerates, D.C. Prodr.

60. Astragalus canescens, So/. MSS.
A, villoso-pubescens, adscendens: caulibus diffusis, adscendentibus :
foliolis multijugis, ovalibus vel oblongo-ellipticis, retusiusculis, supra

of Madera and Porto Santo. 35

glabris, infra hirtis, canescentibus: pedunculis elongatis, folio multim
longioribus: racemis multifloris: pedicellis fructiferis deflexis: legu-
minibus faleatis, compressis, dorso canaliculatis, apice acutis, pubescen-
tibus, pendulis; sulci dorsalis lati, profundi marginibus acutis.

Astragalus canescens, Sol. MSS. et Herb. Banks!

Hab. in Insulé Portu 8S". etiam Canariis ab amico P. B. Webb
arm. lectus.

Flores pallidé flavi, virescentes. 4. hamoso proxima; nec forsan
vere distincta.

Orv. XIX. HYPERICINE®.
Gren. HYPERICUM, D.C.

61. Hypericum angustifolium, Prodr. MS.

HT. glabrum: caulibus simplicibus erectis, strictis, virgatis, anci-
pitibus, suffrutescentibus: foliis epunctatis, erectis, lineari-oblongis, \ob-
tusissimis vel retusis, amplexicaulibus, margine revolutis: panicula ter-
minali, corymbosa: sepalis ovatis, xqualibus, dentato-glandulosis peta-
lisque nigro-punctatis: floribus trigynis.........

Hab. in Madere campo precelso (5000—6000 ped. alt.) “Paul da
Serra” dicto.

Caules plures, feré pedales, tenues.

Orn. XX. MALVACE.

Gen. SIDA, Cav., D.C.
Sect. MALVINDA, Med., D. C.

* * Oblongifolie ; nempe pedicellis elongatis, distinctits articulatis, foliis ob-
longis ovatisve. D.C. Prodr.
62. Sida maderensis, Prodr. MS.

S. fruticulosa: foliis lanceolatis oblongisve, acutis, serratis, glabris,
subtis pallidis, sub-glaucis, breviter petiolatis: axillis inermibus: pedi-
cellis axillaribus, unifloris, inaqualibus, folio brevioribus: carpellis

10—12, uni-rostratis,
EQ

36 Mr. Lowe on the New Plants and Land Mollusca

Malvinda unicornis folio rhomboide perennis, Déllen. Hort, Elth.
p. 216, ¢. 172. f. 212. (descr. et fig. opt.)

Hab. sects vias in locis incultis &c. Madera; in regione tota in-
feriore vulgatiss:

Pedicellis nunquam “ folii longitudine” et carpellis pluribus,
semper uni-rostratis, & .S. canariensi differt. Flores parvi, ochracei.
Fruticulus.

Osp. XXI. VIOLARIE.
Gren. VIOLA, Tourn., D. C.
Sect. I. Nomimiwm, Ging.—

§. 2. ——*, D. C. Prodr. I. p. 295.

63. Viola maderensis, Prodr. MS.

V. caulesceus, stolonifera: caulibus brevibus, erectis, suffrutescen-
tibus, glabris: foliis profundé cordatis, rotundato-ovatis, sub-pubescen-
tibus; petiolis elongatis pedunculisque pube deflexa hirsutis; stipulis
glabris, acuminatis, glandulis ciliato-serrulatis: sepalis oblongis, acutis:
petalis lateralibus vix sub-barbatis: caleare sub-compresso, saccato, ple-
rumque obtusissimo, (rard acuminato): stigmatis rostro uncinato, de-
orstum (se. ad flexuram) immarginato, nudo, complanato (nec convexi-
usculo), styloque compresso, simplici, glabro: capsulis pubescentibus,
hexagonis, globosis, abbreviatis: seminibus albidis, pallidé flavescentibus,
obovatis.

Hab. in Madere sylvis, ubique vulgatissima.

Flores odoratissimi, violacei, sub-pallidiores quam in V.. odoratd.

Orv. XXII. CRUCIFERZ.
Gren. SINAPIDENDRON, 2nob.: Prodr. MS.

Srnarpis, Sect.? 5 Disaccium, D. C
Srinapis, Brown, in Hort. Kew., Hook.
HespPeEris, Spr.
Calyx clausus, demim erecto-patens; basi sub-bisaccatus.
Stylus distinctus. Séigma capitatum. Siliqua linearis, teretius-

of Madera and Porto Santo. 37

cula, sub-torulosa, flexuoso, rostrata, basi tetragona; septo sub-
spongioso. Semina uniseriata, oblonga. Cotyledones incum-
bentes, sub-conduplicate.

Suffrutices Maderenses. Folia sub-carnosa, rigida, simplicia. Flores
flavi, inodori. Silique graciles, elongate, pedicellate. Genus habitu.

calyce, siliquis, seminibusque 4 Sinapi distinctum.
SPECIES.

64. Sinapidendron frutescens, Prodr. MS.

Sinapis frutescens, Ait: Hort. Kew: IV. p.127. n°. 11.—Herb.
Banks! D.C. Prodr, I. p. 220. n°. 34.—Hook: Mise. Bot. 1. p. 119.
Z. 25)!

Hesperis diffusa, Spr. Syst. II. p. 900. n°. 18.

Hab. in rupibus Madere.

65. Sinapidendron salicifolium, Prodr. IS.

S. “caule frutescente; foliis lineari-lanceolatis, integerrimis.” So/.
MSS.

Brassica frutescens, Sol. MLSS. et Herb. Banks!

“Hab. in Madera inter rupes maritimas prope vicum Camara de
Lobos. 4. Fr. Masson.” Sol. MSS.

Species videtur a S. frutescente distincta. Folia succulenta, con-
ferta, integerrima, sub-obtusa, 2—3 poll. longa, 4 poll. lata. Calya semi-
patens. Siligue 1—1} poll. (absque rostro) longz, lineares, flexuosa,
graciles, 4-angulares, longitudinalitér sub-striate ; rostro 4—+4 poll. longo,
capitato, subulato, sub-compresso coronatz.

Habitus omnind S. frutescentis. Plantam vivam nondum vidi:
descriptio 4 specimine Banksiano composita est.

66. Sinapidendron rupestre, Prodr. MS.

S. caule basi frutescente petiolis, foliisque crassiusculis, strigoso-
hispidis; superioribus elongatis, oblongo-linearibus, integerrimis; infe-
rioribus ovato-oblongis, sinuato-dentatis, basi sub-lyratis, petiolatis: sili-
quis glabris; rostro ancipiti, brevi,

38 Mr. Lowe on the New Plants and Land Mollusca

a. chetocalyx; pedicellis, calycibus maculatis, germinibusque hispidis.
Hab. in rupibus convallium Madere.

B. gymnocalyx; siliquis sub-abbreviatis, pedicellis, calycibus sub-imma-
culatis, germinibusque glabris: foliis lucidis; inferioribus rotundatis
obtusis, setis raris scabris.

Hab. in rupe quadam excels’ maritima, ad locum ore Septentri-
onalis Madera “ Entroza” dictum: semel tantim legi.

In a, Calyx purpureo-nigro maculatus. Flores majusculi; petalo-
rum limbo citrino; ungue purpureo.

8. An species? sed habitu eodem gaudet; nec in ceteris charac-
teribus, floribus &e, preter supra indicata, differt.

Orv. XXIII. RANUNCULACE®, D.C.

Gren. RANUNCULUS, C. Bauh., D.C.
OO: Il. Ranuneulastrum. D. C.

67. Ranunculus grandifolius, Prodr. MS.

R. foliis amplissimis, lucidis, cauleque hirsutiusculis; radicalibus
petiolatis, orbiculato-reniformibus, latis, sub-quinquelobis, dentatis; lobis
abbreviatis, rotundatis: caule elato, ramoso, corymboso; ramis divari-
catis, sub-patentibus: corymbo vasto, amplo: calyce patentissimo.

Hab. in rupibus humidis umbrosis Madera; presertim Convallis
frigid (Ribeiro Frio dictz).

Charaeteribus difficillimé, habitu se. staturé, toto ecelo ab affinibus
R. cretico et R. cortusefolio dignoscitur. Plantam in horto cultam
nec solo, nee ccelo aridiore mutatam inveni. Folia radicalia sub-indi-
visa, diametro feré pedali. Caulis 2—3-pedalis. Flores conspicui,
flavi, magni; petalis sc, 1 poll. longis.

of Madera and Porto Santo. 39

MOLLUSCA.

Crass: GASTEROPODA.

Orv. PULMONEA.
1. Familia, Limacide.

I. Genus ARION, Fer.

1. Arion empiricorum, Fer.
a. Varietatis a. Fer. sub-varietates due; altera olivacea vel fusco-
lutescens; altera pallidior, czruleo-cinerascens.
o | Her.
Hab. in Madera.

II. Genus, LIMAX. Fer.
2. Limax antiquorum. Fer.
a. Fer. sub-varietates.
yn. Fer.?
Hab. in Madera.
8. Limax variegatus, 8. Fer.
Hab. in Madera.
4. Limax agrestis, Fer.
e. Fer.
y. Fer.
Hab. in Madera.

40 Mr. Lowe on the New Plants and Land Mollusce

III. Genus, TESTACELLUS, Cuwv.
Testacellus haliotideus, Drap., Sow., Fer.
Hab. in Madera.
6. Testacellus Maugei, Fer., Sow.
Hab. in Madera.

2. Familia, Helicide.
IV. Genus, VITRINA, Drap. <

He.icouimax, Fer.
7. Vitrina Lamarckii, nob. in Zool. Journ.—Tab. 5. ff. 1, a, b.

Helicolimax Lamarckii, Fer.
Hab. in Madera et Portu 8”.

V. Genus, HELIX, Fer. (excluso Sub-genere Cochlodina,
i.e. Clausilia),
Oss. Methodum cl. Ferussaci hine usque ad finem Cochlodontium

sequor.
§§. Incluseze.

+. Volutate, Helicoides.

I. Sub-genus. HELicocrna.

1. Columellate; columella solida torta; globose.

8. Helix furva, Prodr. MS.—Tab. 5. f. 2.

H. testa imperforata, sub-globosa, tenui, fusco 1-fasciata; epider-
mide umbrino: anfractibus obsoleté rugulosis, primo carinato, ceteris
planiusculis: sutura distincté: spira depressiuscula, obtusa: peristomate
simplici, acuto.

Axis 5 lin. Diam. 93. Anfr. 6.

a. fascia continua.
B. fascia interrupta.
Hab. in Madere sylvis; rarior.
9. Helix erubescens, Prodr. MS.
H. testa imperforata, globosa, tenui, rubescente: anfractibus strio-

of Madera and Porto Santo. 41

lis rugisve valdé obliquis, sub-undulatis vel anastomosantibus corru-
gatis; primo vix sub-carinato; ceteris convexiusculis, aquis: spira
elevato-obtusa: peristomate acuto, sub-reflexo, intus sub-incrassato,
carneo.

Axis 4 lin. Diam. 7. Anfr. 5.
a. testa fasciis maculisve fuscis ornatéa.—Tab. 5. f. 3.
3. testa immaculata, unicolore.

Hab. in Madere sylvaticis.
10, Helix sub-plicata, Sow.—Tab. 5. f. 4.

Sow. in Zool. Journ. 1. p. 56. n’ 1. t. i. f 1! (testa decorticata,
semi-fossilis.)
Hab. in Insulé quadam parva, juxta Portum Sanctum, “TIlheo de
Baxo” dicta.
2. Imperforate (Depresse).

Testa depressa umbilicata ; umbilico omnino tecto. Fer.

11. Helix undata, Prodr. MS—Tab. 5. f. 5.

H. testa juniore umbilicata, adulta imperforata, sub-globoso-depressa,
unicolore, fusco-nigrescente : anfractibus corrugatis vel undato-rugosis,
nitidiusculis; ultimo depresso, supra planiusculo; ceteris convexiusculis,
sutura distincta: spira brevi, obtusa, sub-depressa: peristomate simpli-
ciusculo, sub-inerassato, vix reflexo, pallido.

Axis } poll. Diam.1. Anfr. 6.

Helix corrugata, Sol. MSS.; nee Gmel. nec. Dillw.

Helix scabra, Wood's Suppl. ¢. viii. f. 62! nec Chemn., nec Lam.
Feruss. &e.

Hab. in Madere sylvis, graminosis montanis, &c., vulgaris.

12. Helix phlebophora, Prodr. MS.—Tab. 5. f. 6.

H. testa juniore umbilicata, hispida; adulta imperforata, sub-glo-
bosa, fusco bifasciata: anfractibus sub-tumidis, striis crebris, equalibus,
transversis, obliquis, sub-flexuosis sculptis; ultimo ad angulum peristo-

matis inferiorem depressiusculo, ventricoso, prominente : spira conoidea,
Vol. LV. Part I. F

42 Mr. Lowe on the New Plants and Land Mollusca

sub-exserta, obtusa ;sutura distincta, ab angulo peristomatis primo valde
obliqua: apertura rotunda; peristomate continuo, simpliciusculo, paul-
lum incrassato; columella expansa, plana, rosea.

Axis 4—43 lin, Diam. 8. Anfr. 54.

Helix nivosa, Sow. in Zool. Journ. 1. p. 56. n°. 3. t. iii. f 3!

Helix exalbida, Wood, Suppl. ¢. viii. f. 81!

Hab. in Insula Portis S$"; ubique vulgatissima.

Nomen alterum imponendum est ob priora (4 testis quibusdam de-
corticatis, ut videtur, orta) speciei prorsus abhorrentia, ideoque difficul-
tatem indagatoribus vel diligentissimis haud levem parantia. Nomen
itaque novum, quodammodo aptius, ¢ duobus incommodis minus esse
malum videtur: tales enim mutationes pessime auctoritatis, nec nisi
gravissimis argumentis probari possunt. In dilemmate verd tali, quis
inter nominis veteris plane falsi et erronei adoptionem, et aptioris quamvis

recentioris usum hzreret ?

II. Subgenus, HeLicopon. (Helicodonta Fer.)

I. Perrsonare.

Peristoma sinuatum et incrassatum, vel reflexum atque dentatum, dentibus,

laminis, plicisve tortuosis anfractiis penultimi partis convex sepe coarctatum.

13. Helix areta, Prodr. MS.—Tab. 5. f. 7.

Hi. testa rotundata, depressa, utrinque planiuscula, carinata, umbi-
lico minimo perforata, solida, crassa, glabra: anfractibus striis crebris,
zqualibus, transverso-obliquis crassiusculis rudibusve sculptis: spira con-
vexo-depressa; sutura distincta, sub-impressa: apertura transversa, ovali,
dente lamellata intis ad ventrem* coarctata; peristomate albo, reflexo,
continuo, «quali.

Axis 1—1} lin. Diam. 2—21. Anfr. 4—41,

Hab. in Madere collibus aridis maritimis.

' * Venter, pars convexa anfractis penultimi, aperturam (in Helicibus) coarctans,

of Madera and Porto Santo. 43

14. Helix fausta, Prodr. MS—Tab. 5. f. 8.

HZ. testa rotundata, carinata, sub-globoso-depressa, supra* convexi-
ore, pilis brevissimis undique scobinato-hispida, leviuscula: spira eleva-
tiuscula, depresso-conoidea: anfractibus planiusculis, obsoletissimé trans-
verse striatis; sutura distincta, impressa: apertura transversa, intis
angustata, exterits dilatata, dente lamellata intts ad ventrem coarctata:
peristomate extra expanso, sub-reflexo, acuto; columellam versus albo,
incrassato, sub-sinuato sc. obsoleté bidentato, reflexo, umbilicum penitis
obtegente.

Axis 13 lin. Diam. 3. Anfr. 54.

Hab. in sylvis Convallis “Boa Ventura” (i. e. Boni Successtis) dicte.
in Maderz ora Septentrionali.

Helici personate cl: Draparnaudi maximé quidem affinis, sed

distinctissima.

15. Helix arridens, Prodr. MS.—Tab. 5. f. 9.

Hi. testa carinata, umbilico parvo perforata, rotundata, depressa,
utrinque sub-planulata, tenui, hispida, leviuscula: spira convexo-
depressa; anfractibus planiusculis, obsoletissimé transversé striatis; sutura
sub-distineta: umbilico spiralif, rotundo: apertura edentula, transversa,
ints angustata et in umbilicum quasi cum rictu paullim producta;
peristomate interrupto, extra simpliciusculo, sub-reflexo; angulum_ver-
sus internum incrassato, sub-sinuato, albo, reflexo, et umbilicum partim
lamina expansa obtegente.

Axis 1} lin. Diam. 3. Anfr. 44—5.

Hab. in Madera.

Characteribus forté artificiosis cum Hezicert1s Hygromanibus con-
socianda; sed affinitas summa cum priore reliquisque Hzzrcoponrraus

* Supra latus quo umbilicus situs est; infra quo spira, respicit.
t Umbilicus spiralis dicitur ubi plus minus anfractis penultimi, antepenultimi, plu-
riumye intis conspiciuntur. Huic opponitur umbilicus cylindricus.

44 Mr. Lowe on the New Plants and Land Mollusca

naturalis, his adnumerare docet. Helici edentule cl: Draparnaudi
proxima; caret autem impressionibus externis plicarum; caret quidem
omninod plicis ipsis ullis: ideoque ab Helice faustd nostra et H. per-
sonatd Drap':, aliisque hujusce sectionis cognatis, nullo modo intervallo
longo separanda est.

III. Subgenus, Hrevicicona.
1. Carocolle. Uwmbilicus tectus.
16. Helix Webbiana, Prodr. MS\—Tab. 5. ff. 10.

HZ, testa adulta imperforata, tenui, nitida, sub-lampadiformi, de-
pressa, carinata, utrinque convexa, corneo-fuscescente; supra ad umbi-
licum virescente, convexiore, oblique tenuiter striata, carinam versus
utrinque sub-impressam, obtusam, suturamve granulis minutis scabra:
spira convexo-depressa, obtusissima; sutura distincta; anfractibus pla-
nulatis, ultimo maximo: apertura transversa, sub-ovali, amplissima,
patula, extra carina angulata; peristomate interrupto, tenui, acuto;
extra valde expanso, patulo; ad columellam sub-incrassato, sub-reflexo.

Axis 3lin. Diam. 9. Anfr. 3—34.

Hab. in montibus Insulz Portis 8".

Amico P. B. Webb, Arm’., Natur indagatori impigro ac _peritis-
simo, speciem pulcherrimam atque rarissimam dico.

2. Vortices. Umbilicus apertus,
17. Helix Bulveriana, Prodr. MiS_—Tab. 5. ff. 11.

HZ, testa rotundato-depressa, hemispherica, rotata, supra planulata,
acutissimé carinata, tenui, nitidiuscula, tota minutissimé et confertim
granulata, fusco-castanea, supra fasciata: spira convexo-depressa, plus
minus elevata, obtusissima; sutura obsoleta; anfractibus planis, xquis,
quasi attritis vel confluentibus, ultimi cariné acutissima, tenui, supra
sulco exarata, limbata; umbilico patulo, spirali, profundo: apertura

rotundato-lunata: peristomate interrupto, ad umbilicum incrassato, re-
flexo.

of Madera and Porto Santo. 45

Axis 3—21 lin. Diam. 7—8. Anfr. 8—7.

Helix Bulverii, Wood, Suppl. t. vii. f- 82!

Hab. in montibus Insule Portis $8".
18. Helix tectiformis, Sow —_Tab. 5. f. 12.

Sow. in Zool. Journ. p. 57. n’. 6. t. it. f 6!

Hab. in insula quadam parvulé “Theo de Baxo” dicta juxta In-
sulam Portis 8".

19. Helix subtilis, Prodr. MS.—Yab. 5. f. 13.

H. testa orbiculari, utrinque depresso-planulata, tenui, unicolore,
pallidé fusco, acuté carinata: spira depressa, sub-planulata; sutura dis-
tincta; anfractibus planulatis, striis transversis, obliquis, tenuibus, plus
minus distinctis, equidistantibus, interstitiisque striolis subtilissimis,
crebris, decussatis: umbilico patulo, magno: apertura transversa, de-
pressa, obliqué lunata; peristomate interrupto, sub-simplici, sub-reflexo.

Axis 1 lin, Diam. 3—4. Anfr. 5.

An Helix lenticula, Feruss. Tabl. Syst, n°. 154?

Hab. in Maderee maritimis.

20, Helix actinophora, Prodr. MS.—Tab. 5. f. 14.

H. testa orbiculata, depressa, supra convexiore sub-turgida, tenul,
unicolore, fusco-rufescente, acuté carinata: spira convexiusculo-depressa,
sub-planulata; sutura distincta; anfractibus planatis, striis creberrimis,
tenuissimis, transversis undulatim laminosis, quibusdam ad carinam su-
turamve in laminas breves, membranaceas, lacinulasve acutas, radiantes
productas, notatis: umbilico spirali, parvo: apertura transversa, rotun-
dato-ovali, sub-lunata; peristomate interrupto, acuto, patulo, reflexo,

Axis 2 lin, Diam. 4. <Anfr. 5.

Hab. in Madere sylvaticis.

IV. Subgenus, HELIcELLA.

Lomastome ; peristoma reflexum,

21. Helix pulchella, Mudd.
Hab. in Madera,

46 Mr. Lowe on the New Plants and Land Mollusca

292. Helix Porto-sanctana, Sow:
a. vulgaris.—Tab. 5. f 15.
Sow: in Zool. Journ. 1. p. 57. n’. 5. t. iit. ff. 5!
Hab. copiosissimé in Portu 8”,
B?2 gigantea—Tab. 5. f. 16.

Hab. in Portu 8”.

An var. 8. species potitis? quamvis enim ab a. vix nisi magni-
tudine duplo feré majore differt, status intermedios nunquam vidi
Var. a. viva ubique copiosissima; #. rarissima nondum nisi statu semi-
fossili, decorticato occurrit.

Aplostome ; peristoma simplex.
* Verticilli.
23. Helix pusilla, Prodr. MS—Tab. 5. f. 17.

HZ. testa rotundato-depressa, ecarinata, tenui, rufescente: spira con-
vexiuscula; sutura distincta, impressa; anfractibus rotundatis, _ striis
transversis, annularibus, elevatis, sub-membranaceis, tenuibus, remotis,
equalibus, plicatis; interstitiis striolis aliis exilissimis tenuissimisque
creberrimis, spiralibus sc. transversas decussantibus, sculptis: umbilico
patulo, spirali, profundo: apertura rotunda, vix lunata se. circuli seg-
mento perparvo dempto; peristomate simplici (tenui, acuto).

Axis $ lin. Diam, 1. Anfr. 4.

Hab. in Madere sylvis.

Obs. Helici pygmee cl: Drap: quoad staturam et habitum
maximeé affinis; species autem revera distinctissima.

* * Hyaline.
94. Helix bifrons, Prodr. MS —Tab. 5. f. 18.

H. testa rotundato-depressa, umbilicata, sub-carinata, tenui, nitida,
concolore, corneo-virescente; supra leviuscula, obsoleté striata; inferneé
striis valde distinctis sculpta: spira convexo-depressa; sutura distincta,
impressa; anfractu ultimo infra carinam, ceterisque anfractibus, striis
costisve transversis, zqualibus, crebris sulcatis: umbilico parvo, cylin-

of Madera and Porto Santo. 47

drico, sub-spirali, profundo: apertura lunata; peristomate simplici,
tenui, acuto, intus albo marginato.

Axis 23—3 lin. Diam. 6—7. Anfr, 7—8,

Hab. in Maderz sylvis,
25. Helix cellaria, Mull.

Helix lucida, Mont.

Hellx nitens, Drap.

Hab. in Madera.

26. Helix erystallina, Mull,
Hab. in, Madera.

Heliomanes ; peristoma marginatum (“ bordé”).

* Testa depressa vel globulosa.

27. Helix paupercula, Prodr. MS.—Tab. 5. f. 19.

HZ, testa rotundata, planata, supra conyexiore, umbilicata, sub-ca-
rinata, solidiuscula, rudi, feré unicolore, anfractu ultimo supra carinam
obsolete fusco unifasciata: spira planata; sutura impressa; anfractibus
rugosis, eroso-scrobiculatis, minutissime elegantissiméque granulatis; ul-
timo ad aperturam constricto: umbilico largo, patulo, spirali, profundo:
apertura rotundata, coarctata; peristomate continuo, eley ato-disjuncto,
annulari, sub-patulo, acuto; labro intts 1—dentato.

Axis 1 lin. Diam. 2—2t, Anfr, 344,

Hab, in Maderz et Portis S*. maritimis.

28. Helix obtecta, Prodr. MS.—Tab. 5. ff. 20, a, b.

HI. testa rotundata, depressa, inferné planata, supra convexa, um-
bilicata, carinata, solidiuscula, rudi, albida, limo vel terra obducta:
spire planulate anfractibus primis concavyis, ceteris prominentibus tur-
gidis; sutura distincta, valde impressa; anfractu ultimo ventricoso,
carina distincta, utrinque sub-exarata vel sulco obsoletissimo expressa ;
omnibus rugosis, eroso-scrobiculatis, minutissimé elegantissiméque gra-

48 Mr. Lowe on the New Plants and Land Mollusca

nulatis: umbilico mediocri, sub-spirali: apertura rotundata; peristomate
continuo, sub-disjuncto, tenui, acuto, sub-expanso, intus incrassato.

Axis 2 lin. Diam. 5. <Anfr, 4}—5.

Hab. in montibus collibusque aridis Portis S". rarior; copiosior in
Insula “Theo de Baxo” dicta.

Przcedentis forsan status, vel varietas tanttim major.

29. Helix dealbata, Prodr. MS.

H. testa rotundato-depressa, utrinque convexa, umbilicata, carinata,
solidiuscula, albida: spira convexo-depressa, sub-conoidea; sutura sub-
distincta; anfractibus sub-planulatis, transversé rugoso-striatis, plerum-
que minute granulatis; ultimo obtuse carinato: umbilico parvo, patulo,
sub-spirali, minime profundo; apertura rotunda, ochracea; peristomate
continuo, reflexo.

Axis 2lin. Diam. 4—44, altera transversa 3—3}. Anfr. 6.

a. granulata; testa granulata—Tab. 5. f. 21.

Hab. in montibus Portis 8".

B. levis; testa egranulata, levi, nitida.

Hab. in insula “Tlheo de Baxo” dicta. Status a, solo calcareo
ortus.

30. Helix maderensis, Wood.—Tab. 5. f. 22.

H. testa rotundato-depressa, utrinque planulata, umbilicata, cari-
nata, solidiuscula; supra leviuscula, fusco 1-fasciata; inferné striata:
spira convexiuscula, sub-planulata; sutura distincta; anfractibus pla-
natis, infra transverse striatis, ultimi ad aperturam granulati carina
acuta, striis supra carinam obsoletis: umbilico lato, patulo, spirali:

apertura rotunda; peristomate continuo, circinato, annulati sub-dis-
juncto, crassiusculo, sub-expanso.

Axis 14 lin., rariss. 2; Diam. 3, rariss. 4. Anfr. 6—7.
Helix maderensis, Wood. Suppl. t. viii. f. 84!
Hab. in Madera; vulgatissima.

31. Helix compar, Prodr. MS.—Tab. 5. f. 23.

HI. testa rotundato-depressa, utrinque planulata, umbilicata, sub-
carinata, solidiuscula, fusco bifasciata, utrinque plicato-costata: spira

of Madera and Porto Santo. 49

conyexiuscula, sub-planulata; sutura distincta, impressa; anfractibus
convexiusculis, plicis vel striis transversis, elevatis, acutis, distinctis,
crebris, zquidistantibus aqualibusque costatis; interstitiis levibus; ul-
timi carina obtusa: umbilico lato, patulo, spirali, profundo: apertura
rotundato-oyali; peristomate continuo, circinato, sub-disjuncto, crassius-
culo, reflexo.

Axis 13 lin. Diam. 34. Anfr. 6.

Hab. in Madere collibus maritimis; rariss.

32. Helix leptosticta, Prodr. MS—Tab. 5. f. 24.

HZ. testa rotundato-depressa, umbilicata, sub-carinata, nitidiuscula,
tenui, pallidé cornea, obsolete fasciata: spira convexo-depressa; sutura
distincta; anfractibus convexis, sub-striatis, minuté et elegantissimé re-
ticulato-granulatis ; ultimi carina obtusa: umbilico patulo, spirali: aper-
tura rotundato-ovali; peristomate continuo, simpliciusculo, sub-incras-
sato, sub-reflexo.

Axis 1} lin. Diam. 3. Anfr. 5—51.

Hab. in Mader collibus maritimis.

33. Helix lentiginosa, Prodr. MS'—Tab. 5. f. 25.

H. testa rotundato-depressa, supra sub-planulata, umbilicata, sub-
carinata, tenui, maculata et sub-fasciata: spira convexo-depressa ; sutura
distincta; anfractibus convexiusculis, striato-scobinatis vel squamuloso-
cancellatis, striis se. interruptis squamiformibus, lunatis, quincuncialibus
sculptis: umbilico mediocri, sub-patulo, spirali: apertura transverse
ovali, sub-lunata ; peristomate interrupto, reflexo.

Axis 14 lin. Diam. 2}—3. Anffy. 5.

Hab. in Madere rupibus maritimis.

Helici arridenti nob: affinis.
34. Helix calva, Prodr. MS.—Tab. 5. f. 26.

H.. testa rotundato-globulosa, sub-depressa, imperforata, vix sub-
carinata, nitidiuscula, sub-tenui, sub-pellucida, obsoletissimé 2—fasciata :

spira convexa, elevatiuscula; sutura distincta; anfractibus planiusculis,
Vol. lV. Part I. G

50 Mr. Lowe on the New Plants and Land Mollusca

transverse costulato-striatis, striolisque spiralibus _obsoletissimis, sub-
tilissimis exilissimisque, «quis notatis; ultimo feré ecarinato, ad aper-
turam ochraceo, supra nitido, levi: umbilico clauso: apertura trans-
versa, multd latiore quam alta, sub-lunata, ints angustata, extrorsum
ampliore; peristomate longé interrupto, incrassato, sub-reflexo.

Axis 2—21 lin. Diam. 34—4. Anfr. 63—7.

Hab. in Madere sylvis.

Helici edentule Drap*. aliquatenis forma affinis; sed vix HELIco-
poNTIBuUs releganda.

Hewiceiiis Hyalinis affinitate naturali, et pra ceteris H. bifronti
nostre accedit; peristomate verd marginato, striisque subtilissimis te-
nuissimisque spiralibus prorsts aliena.

35. Helix abjecta, Prodr. MS—Tab. 6. f. 1.

H. testa parvula, rotundato-pyramidata, conoidea, carinata, umbili-
cata, crassa, solida, rudiuscula, utrinque scabra vel granulosa, rugosa,
supra carinam fusco pallidé unifasciata: spira convexa, conoidea; sutura
distincta; anfractibus compactis, convexiusculis, transversé rugosis et
granulatis: ultimi carina sub-acuta, ad suturam approximata: umbilico
parvo, spirali, profundo: apertura rotundata; peristomate continuo,
reflexo.

Axis 13—2 lin. Diam. 3—3}. Anfr. 64—7.

Hab. in insula Portas S"., una cum H. compacta degens; vulga-
tissima.

Inter H. compactam et H. echinulatam nostram quasi intermedia ;
ab utraque satis distincta: priori verd quam maximé affinis. .

36. Helix compacta, Prodr. MS.—Tab. 6. f. 2.

H. testa parvula, rotundato-globulosa, sub-conoidea, perforata, sub-
carinata, crassa, solida, rudiuscula; infra’ scabra, rugosa; supra leviore,
nitidiuscula, pallidiore, fusco obsoleté 1-fasciata: spira convexa, eleva-
tiuscula; sutura distincta; anfractibus compactis, planiusculis, transverse
striatis et granulatis; ultimo sub-carinato, supra angulum leviusculo

of Madera and Porto Santo. 51

se. egranulato, umbilicum coarctante: umbilico minimo, sub-spirali,
rimaformi: apertura rotundato-lunata; peristomate interrupto, (labris
approximatis, aliquando sub-continuis) sub-reflexo,

Axis 2 lin. Diam. 3. Anfr. 6—64.
Sowerb. in Zool. Journ. I. ¢. iii. f. 8!

Hab. in Insula Portis S". gregaria, ubique copiosissima: in Ma-
dera ad Promontorium S". Laurentii (“Ponta Sad Lourenco”) soliim.

37. Helix consors, Prodr. MS.—Tab. 6. f. 3.

H. testa rotundato-depressa, perforata, vix sub-carinata, crassius-
cula, solida, rudiuscula; infra preesertim scabra, rugosa; supra leviore;
utrinque pallido fuscoque variata, ad aperturum ochracea: spira con-
vexo-depressa; sutura sub-indistincta; anfractibus planatis, transverse
striatis et granulatis; ultimo sub-carinato, umbilicum coarctante, gra-
nulis supra angulum obsoletis: umbilico minimo, sub-spirali, rimzformi :
apertura rotundato-lunata; peristomate distincté interrupto, sub-reflexo.

Axis 24—3 lin. Diam. 44—5. Anfr. 6—61.
Hab. in Insula Portas S". cum precedente; rarior.

Precedenti vel maximé affinis; characteres itaque extricatu diffi-
cillime: sed forma magis depressa numerusque anfractuum isdem,
quamvis testa feré duplo major, speciem esse distinctam suadent; ob-
stante nulla differentia loci, soli, cibi, nee alia quapiam hujusmodi
causé que talem mutationem efficere posset.

38. Helix depauperata, Prodr. MS.—Tab. 6. f. 4.

H. testa rotundato-depressa, umbilicata, ecarinata, tenuiuscula, su-
pra convexa, unicolore, sordida, minutissimé et elegantissimé confertim
reticulato-granulata : spira convexo-depressa; sutura distincta, sub-
impressa; anfractibus convexis, sub-tumidulis, transversé sub-striatis ;
ultimo sub-rotundato: umbilico mediocri, aperto, spirali, profundo:

52 Mr. Lowe on the New Plants and Land Mollusca

apertura rotundata; peristomate sub-continuo, simpliciusculo, tenui,
intus sub-marginato.

Axis 2—21 lin. Diam. 4-43. Anfr. 5—5}.

Hab. in montibus Insule Portis S$".

39. Helix lurida, Prodr. MS.—Tab. 6. f. 5.

HZ. testa sub-globosa, depressiuscula, supra convexa, umbilicata,
ecarinata, tenuiuscula, fusco sub-fasciata, nitidiuscula: spira conyexo-
depressa; sutura distincta; anfractibus convexis, minutissimé et ob-
soletissimé confertim reticulato-granulatis; ultimo rotundato, juxta
suturam granulato, superné levi sc. egranulato: umbilico parvo, cylin-
drico, profundo, aperto: apertura lunata, sub-ovali; peristomate sim-
plici.

Axis 3 lin. Diam. 5. Anfr. 54—6.

Hab. in montibus Insule Portis S*.

Sequenti proxima.
40. Helix nitidiuscula, Sow—Tab. 6. f. 6.
Sow. in Zool. Journ. I. Poin, Ac bil. fey
Hab. in Madera et Portu S°; ubique vulgatissima.
41. Helix punctulata, Prodr. MS.—Tab. 6. ff. 7, 8.
. setulosa; testa sub-tenui, sub-inflata, scabra, spinelloso-hispida.
Axis 4 poll. Diam, 3. Anfr. 5.—Tab. 6. f. 7.
Helix punctulata, Sow. in Zool. Journ. 1. p. 56. n°. 2. ¢. iti. f. 2!
B. solida; testa solida, glabriuscula, pallida.
Axis 3 poll. Diam. 3. Anfr. 5.—Tab. 6. f. 8.
Hab. in Portu 8»,
An satis ab. H. nitidiusculd Sow. distincta ?
42. Helix pisana, Mull.
H. rhodostoma Drap., cmgenda Mont., &c.

of Madera and Porto Santa. 53

Hab. in Mader Promontorio “P". Sad Lourenco” dicto. In

Portu 8S”. vinetarum calamitas.

43. Helix lauta, Prodr. MS\—Tab. 6. f. 9.

H. testa sub-globosa, supra convexa, umbilicata, ecarinata, tenui-
uscula, (alba, fasciis angustis, interruptis, fuscis, obsoletis ornata), niti-
diuscula: spira convexo-depressa, sub-elevata; sutura distincta; anfrac-
tibus convexis, striis confertissimis, equalibus, concinnis, transversis
sculptis; ultimo rotundato: umbilico parvo, cylindrico, profundo,
aperto: apertura lunata, sub-rotunda; peristomate acuto, intts annulo
distincto, elevato, margini approximato.

Axis } poll. Diam: %. Anfr. 5.

Hab. in Portu 8".

Specimen unicum decorticatum tantim habeo, a Rev’. Dom’.

Bulwer repertum, quod mihi cl. G. B. Sowerby humanissimé commu-
nicavit. H. luride nostre, necnon H. striate Drap: (caperate Mont :)
et forsan aliis quibusdam proxima: sed ab omnibus distincta videtur.

44, Helix striata, Drap*.?

Hab. in Madera; rariss.

Differt umbilico et numero anfractuum, pro magnitudine, majore.
Quum autem testas perpaucas easque nondum adultas adhuc repertas

habeo, distinguere vix audeo.

45. Helix rotula, Prodr. MS.—Tab. 6. f. 10.

Hi. testa rotundata, conoideo-depressa, supra sub-planulata, sub-
perforata, carinata, scabra, nitidiuscula, fasciata: spira conoidea, obtu-
sissima; sutura obsoleta; anfractibus planis, transverse striatis et granu-
latis; ultimo acute carinato, carina ad peristoma obsoleta: apertura
lunata, extrorsum dilatata; peristomate intis incrassato, acuto, sub-ex-
panso; ad angulum internum reflexo, calloso, perforationem obtegente.

Axis 3 lin. Diam. 6. Anfr. 8

Hab. in montibus Porttis S*.

54 Mr. Lowe on the New Plants and Land Mollusca

46. Helix polymorpha, Prodr. MS.—Tab. 6. ff. 11—16.

HZ testa rotundato-depressiuscula, umbilicata, carinata, crassiuscula,
solida, fusco fasciata et maculata: spira conoideo-depressa, aliquando
feré planata, granulata; anfractibus planiusculis; primorum saltem su-
tura obsoleta; ultimi carina plis minis acuta: umbilico patulo, spi-
rali, largiusculo: apertura lunato-rotundata; peristomate sub-reflexo.

a. irrasa; testa depresso-conoidea, sub-globulosa, utrinque granulato-
scaberrima, limo vel terra obducta: spira convexo-elevatiuscula,
conoidea; anfractibus convexis; sutura distincta; carina obtusa:
peristomate sub-interrupto.

Axis 3 lin. Diam. 5. Anfr. 8.

Albida, fasciis fuscis distinctis, superiore lato, continuo, distinc-
tissimo; infra sc. spira sub-maculata, variegata.

Pn ive

Hab. in solo rubro “Tufa” Geologicis dicto, ad promontoriam
St, Laurentii Madere.

B. depressiuscula; testa rotundato-depressiuscula, obsoleté utrinque gra-
nulata, supra presertim nitidiuscula, leviuscula sc. granulis raris,
obsoletis: spira convexo-depressiuscula; anfractibus convexiusculis ;
sutura distincta; carina obtusa: peristomate interrupto.

Axis 24 lin. Diam. 5—53. Anfr. 7.

Supra albida, fasciis fuscis, superiore lato, continuo, distincto, ce-
teris interruptis vel obsoletis; infra sc. spira albido fuscoque maculata,
variegata.

ey IED res

Hab. in solo Tufa dicto in collibus maritimis prope urbem Fun-
chalensem Madere.

y- arenicola; testa rotundata, supra sub-planulata, utrinque granulata ;
supra presertim nitida, granulis obsoletioribus: spira convexo-

of Madera and Porto Santo. 55

depressiuscula, plis minus elevata; anfractibus convexiusculis; su-
tura distincta; carina sub-obtusa: peristomate sub-continuo.

Axis 2—21 lin. Diam. 44—5. Anfr. 7.

Sub-var. 1. Supra fusco fasciata; fascia superiore distincta, sub-continua,
angusta; ceteris interruptis.

f. 13, l. ¢.
2. Inornata se. non fasciata, variegata.
Albida, nitida, spira fusco maculata, variegata. Colores quodam-
modo letiores quam in ceteris; albo prasertim clariore.
Status a, é solo ealcareo ortus.

Hab. in arenosis calcareis Promontorii S*. Laurentii Madere.

8. attrita; testa rotundato-depressa, rotata; infra planulata; supra con-
vexa, nitidiuscula; utrinque confertim granulata: spira convexo-
planata; anfractibus planis, quasi attritis; sutura obsoletissima ;
carina acutissima: umbilici margine (presertim in junioribus)
abrupto, declivi: peristomate feré interrupto.

Axis 2 lin. Diam. 44—51. Anfr. 7.
Sub-var. 1. Supra pallida, fusco fasciata; fascia superiore angusta; ple-
rumque unica.
f. 14, l. ec.
Helix tectiformis, Wood. Suppl. t. viii. f. 83!

2. Tota fusca, sub-unicolor, preter spatium vel fasciam latam
pallidam cirea umbilicum.

3. Tota variegata, nec fasciata.
Sub-varietas quaque colore magis fusco quam in ceteris gaudet:

in 1™ sordidé albido vel pallidé ochraceo fuscoque variegata et macu-

lata; anfractis ultimi pars semper in omnibus juxta peristoma ochracea,
immaculata.

Hab. in collibus montibusve Portis S".

56 Mr. Lowe on the New Plants and Land Mollusca

«. calcigena; testa rotundato-depressa; supra planulata, levi, nitida, ad
aperturam tantiim sub-granulata: spira convexo-depressa, pltis mi-
nus elevata, granulata; anfractibus planatis; ultimi sutura im-
pressa, ceterorum obsoleta; carina sub-acuta: peristomate sub-
interrupto.

Axis 2—21 lin. Diam. 5—5$. Anfr. 73.
Sub-var. 1. Supra tota alba; spira albida fusco variegata.
ii Tish lero!
2. Supra fasciata; spira albida fusco variegata.
Status 4. vel ¢., solo calcareo ortus.
Hab. in solo calcareo Insule cujusdam, ‘“ Baxo” dicta, juxta Por-

tum S™”,

¢. pulvinata; testa rotundato-conoidea, utrinque confertim granulata :
spira elevata, conica, anfractui ultimo quasi superimposita; anfrac-
tibus (preter primos) convexiusculis, ultimo tumidulo; sutura dis-
tincta, impressa; carina sub-obtusa: peristomate continuo.

Axis 21—3 lin. Diam. 5. Anfr. 73.

Sub-var. 1. Supra tota alba spira sub-maculata.
2. Supra fusco fasciata; spira maculata—t. 16, l. c.

Colores in utroque statu (sc. sub-varietate) quam in ceteris varie-
tatibus longé pallidiores. Testa quidem in omnibus pallida, albida,
apice spire fusco.

Hab. in montibus collibusve Portis S“.; cum 36. aétrita nostra de-
gens.

Varietates 6. et ¢ (forsan etiam e) primo aspectu distinctissime, tot
forsan species constituende quibusdam videantur. Aded tamen, medi-
ante ¢, sunt conjunctae, ut tres ille 4, «, ¢ nee a seipsis nec ab a, B,
y, quibus ordine inverso analog sunt, separari debent. Sed in re tam
dubia, non is sum qui cuilibet meas varietates pro speciebus habenti,
increparem.

* * Testa trochoidea, carinata.

of Madera and Porto Santo. 57
47. Helix cheiranthicola, Prodr. MS.—Tab. 6. f. 17.

HI. testa pyramidata, conoidea, umbilicata, carinata, solidiuscula,
tota scabra, plerumque fasciata: spira elevata, pyramidata, obtusa; su-
tura distinctissima, impressa; anfractibus convexis, tumidis, distinctis,
confertim granulatis; ultimi carina obtusa: umbilico mediocri, patulo,
spirali profundo: apertura rotundata; peristomate continuo, sub-dis-
juncto se. circinato, incrassato, sub-reflexo.

Axis 3 lin. Diam. 4. Anfr. 8.

Sub-var. 1. xonata; supra fasciata: spira fascia, unica, lata, juxta sutu-
ram: carina albida.

ie Jeri 1G.

2. maculata; supra fasciata: spira maculata vel variegata, nec
fasciata.
3. albida; tota albida, nee fasciata: spira sub-maculata.

Hab. in arbusculis Cheranthi tenuifolii Herit: in monte Portis
S". quodam “ Pico branco” dicto: et in Insula “Ilheo de Baxo” dicto,
sed rarissima.

Varietati ¢. pulvinate Helicis polymorphe nimis forsan affinis: sed
forma et anfractibus tumidis et sutura impressa dignoscitur.

48. Helix oxytropis, Prodr. MS.—Tab. 6. f. 18.

H. testa depresso-conoidea, supra planulata, perforata, carinata,
tota scabra, fusca, sub-fasciata: ‘spira depresso-conica; sutura distincta ;
anfractibus planiusculis; ultimi carina acuta, distinctissima, supra mar-
ginata sc. exarata vel suleo expressa; omnibus distinctissimé confertim
granulatis, asperis: umbilico minimo, sub-spirali, aperto: apertura ro-
tundata; peristomate continuo, circinato, disjuncto, reflexo.

Axis 21 lin. Diam. 4. Anfr. 64.

Hab. in collibus maritimis Portis S*.

49. Helix echinulata, Prodr. MS'—Tab. 6. f. 19.
H. testa parvula, conoidea, sub-pyramidata, depressiuscula, supra

planulata, perforata, carinata, tota scaberrima, fusca, supra fasciata:
Vol. IV. Part. I. H

58 Mr. Lowe on the New Plants and Land Mollusca

spira pyramidata elevata; sutura distincta, impressa; anfractibus con-
vexis; ultimi carina acuta, distincta, supra marginata sc. sulco ex-
pressa vel exarata; omnibus granulis distinctissimis, confertis, asperri-
mis scobinatis et quasi echinulatis: umbilico parvo, sub-spirali, aperto;
apertura rotundata; peristomate continuo, circinato, disjuncto, reflexo.

Axis 2 lin. Diam. 2}. Anfr. 6.

Hab. in monte “ Pico branco” dicto Insule Portis S".

Species elegantissima.

50. Helix duplicata, Prodr. MS.—Tab. 6. f. 20.

Helix bicarinata, Sow. in Zool. Journ. 1. p. 58. n°. 7. t. iti. f. 7!
Wood, Suppl. t. viii. f. 85! non Feruss.

Monstrosa; anfractu ultimo disjuncto; sutura profunda, excavata.

Hab. in Insulé Portis S“.

Nomen egré, et quasi coactus, mutavi; ob Helicem C. bicarina-
tam cl: Ferrussaci, Tabl. Syst. n°. 350.

51. Helix turricula, Prodr. MiS.—Tab. 6. f. 21.

Hi. testa turrita, pyramidata, sub-cylindrica, bicarinata, perforata,
tota minuté et confertissimé granulata, fusca, feré unicolore, vel supra
obsoleté fasciata: spira valde elevata, obtusissima; sutura distincta ;
anfractibus bicarinatis, carinis equalibus, prominentibus, distinctis, sulco
divisis: apertura rotunda; peristomate continuo, circinato, disjuncto,
tenui, reflexo.

Axis 4 lin. Diam. 3. Anfr. 8—84.

Hab. in Insulé quadam “Tlheo de Cima” dicta, juxta Insulam
Portum S™™.

Species notabilior, elegans.
52. Helix bicolor, Prodr. MS.—Tab. 6. f. 22.

H. testa globuloso-conoidea, sub-imperforata, leviuscula, nitida, vix
sub-carinata, fasciis albis fuscisque lete-coloribus ornata: spira_elevati-
uscula, obtusissima; sutura distincta; anfractibus sub-planulatis, trans-

of Madera and Porto Santo. 59

versé striatis: apertura extrorsim ampliore; peristomate longé inter-
rupto, tenui, simpliciusculo, inttis ad angulum incrassato, reflexo, per-
forationem minimam feré obtegente.

Axis 2 lin. Diam. 3. Anfr. 7.

Hab. in summo cacumine montis “ Pico de Facho” dicto Porttis S“.

Species nitidissima, coloribus distinctissimis sc. fasciis laeté coloratis
gaudens. Ob affinitatem Helict maritime Drap., cel: Ferussaco obse-
cutus, hue relegavi; sed ambe potits priori sectioni post Helicem

variabilem Drap. (H. virgatam, Mont.) inserende sunt.

+ Evolutate, Cochloides.
* Apertura feré edentula.

1. Columella solida.
——, planata et ad basin _truncata.

5. Sub-genus, Cochlicopa.

Styloides; testa turrita, apertura brevi, &e.

53. Helix C. acicula, Fer.

Bueccinum Acicula, Mul/—Buce. terrestre, Mont.—Bulimus Acicula.
Brug. et Drap.—Achatina Acicula, Lam* et Nils.
Hab. in Madera.

54. Helix C. tornatellina, Prodr. MS.—Tab. 6. f. 23.

HZ. testa obovato-oblonga vel obconico-cylindrica, levi, nitida, cor-
neo-rufescente (castanea): spira breviuscula, obtusa, duas partes ex
quinque totius longitudinis wquante; anfractibus planis; sutura obso-
leta: apertura longitudinali, coarctata, postice valde angustata; labro
anticé producto, porrecto, sub-inflexo, posticé sub-sinuato: columella
prominula, abrupté et oblique truncata, torta; plica in ventrem lon-
gitudinali, sub-obsoleta, callosa, labro adversa, ad partem posticam an-
gustatam aperture, hane coarctante.

60 Mr. Lowe on the New Plants and Land Mollusca

Long. 4—5 lin. Diam. 2—27. Anfr. 7.
Spira
Apertura’
Hab. in Madera.
Hee et 3 forsan sequentes Helici folliculo affines.
55. Helix C. melampoides, Prodr. MiS.—Tab. 6. f. 24.

TE. atest; Mbovate-oplangay eae. es, LORE SS

2
3°

2
3

wae ee wal eae 20s 25000). 2 spira” breviuseula, obtusissima, partes
duas fereé ex quinque totius longitudinis zquante; anfractibus planis,
ultimo sub-ventricoso; sutura obsoleta: apertura longitudinali, anticé
effusa, sub-patula, omnind edentula; labro recto, equali: columella ob-
soleta, obliqué truncata.

Long. 54 lin. Diam. 2}. Anfr. 6.

Spiral 4
Apertura’ *

Hab. in Insulé quadam, Portum Sanctum ab oriente spectante,.
“Tlheo de Cima” dicta. v. m.

Priori nimis affinis, et forsan varietas tanttm; at major, aliquan-
tulum feré ventricosior, apertura semper edentula, anticé magis effusa,
posticé minus angustata, columella obsoletiore, et labro recto, quali,
nec sinuato, nec anticé producto. Testa decorticata, crassa, solida,
opaca; sed hee etiam in priore (H. éornatellina), post mortem animalis
obtinent: vivam nondum vidi.

56. Helix C. tritieea, Prodr. MS.—Tab. 6. ff. 25, 26.

H. testa obovato-cylindrica, sub-gracili, sub-conica, nitida, levi:
spira acutiuscula, dimidium teste aquante; anfractibus planis; sutura
obsoletiuscula: apertura obovata, biplicata; plica altera transversa, inter
columellam. et angulum labri in medio posita, altera magis interna mi-
nore in columellam; duabus aliquando obsoletis; columella anticé lata,

sub-expansa, plana, vix truncata, in labrum simplex rectum equale atte-
nuata.

of Madera and Porto Santo. 61

Long. 3 lin. Diam. 14. Anfr. 6.
a. biplicata; apertura 2—plicata—f. 25, 1. ¢.
B. edentula; plicis obsoletis.—f. 26, 1. e.
Hab. in Portu S*.

57. Helix C. ovuliformis, Prodr. MS.—Tab. 6. f. 27.

H. testa angusto-elliptica, sub-pupeformi, diametro utrinque zquali,
abbreviata, nitida, levi: spira obtusissima, dimidium teste aequante ;
anfractibus convexiusculis, sub-tumidis; sutura distincta: apertura ob-
ovata, angusta, biplicata; plica altera transversa, abrupta, prominente,
inter columellam et angulum labri in medio posita; altera ad colu-
mellam, magis obsoleta, obliqua: columella expansa, tenui, torta, obli-
que truncata.

Long. 2 lin. Diam. 1. Anfr. 4,

Hab. in cacumine montis “Pico de Facho” in Insula Portis S*.

58. Helix C. gracilis, Prodr. MS.—Tab. 6. f. 28.

HI. testa elongato-obovata, gracili, tenui, vitrea, nitida, levi, (im-
perforata): spira sub-attenuata, obtusa, dimidium teste excedente; an-
fractibus planiusculis; sutura obsoletiuscula: apertura obovata, eden-
tula: columella lata, expansa, vix truncata, in labrum tenue, sub-mar-
ginatum attenuata.

Long. 2 lin. Diam. 1. Anfy. 5,

Hab. in monte “Pico Branco” Insule Portis St“.

Facies Helicis (Cochlicelle) Clavuli Fer. (H. Goodalli, Mill. An-
nals of Philos.); sed magis turrita; anfractu ultimo cum penultimo
majore; sutura obsoletiuscula, non distincta, impressa; testa lavissima,
imperforata, nec striata, nec sub-perforata. Inter Helicem triticeam et
Helicis lubrice varietatem nostram quodammodo media, ab utraque
distincta.

59. Helix C. lubrica, Mudl.—Tab. 6. f. 29.

Var. testa aperturaque angustiore, minus ventricosa, magis elongata.
Hab. in Madera.

62 Mr. Lowe on the New Plants and Land Mollusca

re)

Testa perforata vel umbilicata &e.; peristomate simplici.
a. Anfractibus zqualibus, ultimo ceteris omnibus breviore.

6. Sub-genus, Cochlicella.
60. Helix C. ventrosa, Fer.
Bulimus ventricosus, Drap.
Hab. in Madera et Portu 8”.
61. Helix C. decollata, Linn.
Bulimus decollatus, Drap.
Hab. in Madera.

* * Apertura feré dentata vel laminata.
1. Ecanaliculate ; peristomate plerumque non continuo.

7. Sub-genus, Cochlodon. (Cochlodonta Fer.)
1. Testa cylindrica.
62. Helix C. anconostoma, Prodr. MS'—Tab. 6. f. 30.

H. testa cylindrica, pupzformi, leviuscula, nitida, corneo-rufes-
cente: spira obtusa; anfractibus convexis, rotundatis, quis, striis
transversis, obliquis, obsoletis, indistinctis; sutura distincta, impressa:
apertura 1-dentata, elliptica, sub-angustata, longiore quam lata, sub-tri-
gona, anticé angulata: columella recta, supra cubito vel flexura
abrupto, acuto, cum labro tenui, reflexo conjuncta: dente lamellato in
ventrem juxta labrum obsoletiusculo, 4 labro distincto.

a, gyrata; testa elongata: aperture cubito distinctissimo. f. 30. 1. ¢.

Long. 13 lin. Diam.1. <Anfr. 7.

B. curta; testa abbreviata: aperture cubito obsoletiore.

Long. 14 lin. Diam. 3. Anfr. 6.

Hab. in Madera.

Helici C. umbilicate, Fer. (Pupa umbilicata, Drap., Lam*.; Turbo
muscorum, Mont. t. xxii. f. 3.) proxima, presertim per varietatem (ra-
riorem) 8. curtam; sed distincta videtur. In Icone cl: Montagui su-

pra indicata, testa Britannica ejusque characteres 4 nostra Maderensi
optimé distinguuntur.

of Madera and Porto Santo. 63

63. Helix C. cheilogona, Prodr. MS.—Tab. 6. f. 31.

Hi. testa sub-ovata, cornea, levi, vel obsoleté striata: apertura
3-plicata, coarctata, anticé prominula; plica unica in columellam;
duabus parallelis in ventrem positis; intermedio minore: labro ex-
panso, ints marginato, sinuato-angulato: umbilico magno, patulo, pro-
fundo.

Long. 12 ln. Diam. 1. Anfr. 6.

Hab. in Madera.

64. Helix C. sphinctostoma, Prodr. MS.—Tab. 6. f. 32.

HT. testa cylindrica, fusca: anfractibus planis transversé sub-stri-
atis: apertura 4—6-plicata; plicis duabus in columellam, postica obso-
leta; duabus in ventrem, quarum anterior plice anteriori columellari
zqualis, posterior magna, complicata, cum dente ad angulum inferiorem
labri posito in unum conjuncta: labro reflexo, posticé sub-angulato vel
sinuato, ad angulum intts dentato, anticé 1-2-plicata : umbilico pa-
tulo, profundo.

Long. 2 lin. Diam. 1 Anfr. 7.

Hab. in Madera.

Testa plus minis striata.

65. Helix C. monticola, Predr. MS.—Tab. 6. f. 33.

HI. testa cylindrica, castanea, pallido fasciata: anfractibus con-
vexis, tumidis, striis elevatis, quidistantibus, transversis sculptis ;
sutura impressa: apertura sub-sexdentata; columella 92-plicata, _plica
posteriore obsoletissima; plicis duabus approximatis, parallelis, in ven-
trem, gquarum anterior minor; posterior magna, cum labro continua:
labro sub-reflexo, wquali, 3—plicata; plica intermedia majore; anteriore
et posteriore minutis.

Long. 13 lin. Diam. vix 1. Anfr. 6.

Hab. in summo cacumine Montis “Pico de Facho” Insule Por-
tus SY.

Oss. Priori (H. sphinctostomati) quoad plicas affinis; sed distincta.

64 Mr. Lowe on the New Plants and Land Mollusca

66. Helix C. calathiscus, Prodr. MS—Tab. 6. f. 34.

HZ. testa cylindrica, ovoidea, abbreviata, castanea, pallido fasciata:
anfractibus convexis, sub-tumidis, costulis aquidistantibus, transversis,
crebris, sculptis; sutura impressa: apertura sub-septemplicata; colu-
mella 1-plicata; plicis duabus in ventrem, quarum anterior valde in-
terna, minuta, dentiformis; altera posterior magna, cum labro continua:
labro expanso, sub-sinuato; callo ints margini parallelo, postic® in
dentem duplicem desinente, anticé dente minuto, simplici, obsolete et
plica unica intermedia, magna, instructo.

Long. 17 lin. Diam. 1. Anfr. 7.

Hab. in summo cacumine montis “ Pico de Facho” Portis S*.

67. Helix C. cassida, Prodr. MS\—Tab. 6. f. 35.

H. testa ovata, ventricosa, abbreviata, sub-imperforata: anfractibus
planis, striis elevatis, crebris, aquidistantibus, transversis; sutura sub-
indistincta: apertura 7—8-plicata; columella biplicata, plica posteriore
minore; plicis duabus sub-zqualibus, parallelis in ventrem positis, ex-
teriore paulld majore, cum labro continua, sinum efficiente: labro ex-
panso, 5-plicata; plica anteriore minore, aliquando obsoleta; tribus in-
termediis lateralibus, approximatis, superiore magna, duabus inferioribus
minoribus, quarum infima sub-dentiformis; quinta infima minima, ad
angulum labri posita: perforatione minima.

Long. 2 lin. Diam. 13. Anfr. 7—8.

Hab. in Maderz convallibus, in rupibus aridis umbrosis.

Recens semel tantum lecta; necdum vivam vidi. Ad locum
“Canical” dictum, inter alias plurimas Helicis species* paulld frequen-
tior, sed statu semifossili. ;

* Teste illa, hie et in Portu S*., statu semi-fossili, in arena calcareA, inter concreta
ramiformia (minimé “ Lignites”), reperte, omnes terrestres, plurime (forsan omnes) etiam
hodié in Madera vel Portu S®. yivunt; nec ullam quidem speciem marinam cum illis com-
mixtam vidi. “ Delphinula sulcata Lam*.?” Bond. Exc. p. 140. f. 33. a, b, est Helicis
species, (Helix Delphinula nob.) valde elegans, Helici tectiformi Sow. affinis. A Delphi-
nul& prorsts aliena.

of Madera and Porto Santo. 65

VI. Genus, CLAUSILIA, Drap’.
Helicis sub-genus Cochlodina, Fer.
68. Clausilia crispa, Prodr. MS.—Tab. 6. f. 36.

C. testa turrita, sub-ventricosa: anfractibus convexiusculis, striis
transversis, creberrimis, minutissimé flexuosis sculptis, interstitiis elegan-
tissimé decussatim punctulato-striatis, quasi cancellatis; sutura distincta,
impressa: apertura oblonga, biplicata; plicis columellaribus, approxi-
matis, divaricatis, sub-posticis, postica prominente sc. extrorsiim ad mar-
ginem peristomatis producta eique continua, sinum ad angulum posti-
cum aperture formante: peristomate simplici, acuto, sub-expanso: costis
dorsalibus rimaque umbilicali obsoletis.

Long. 7 lin. Diam. 2. Anfr. 9.

Hab. in rupibus sylvarum Madere.

Peristomate nec elevato neque disjuncto sc. columellari obsoleto,
necnon costis duabus dorsalibus rimaque umbilicali sub-nullis, quin et
quodammodo forma et magnitudine ad Clausiliam bidentem Drap.
(Turbinem laminatum Mont.) magis quam ad aliam quampiam speciem
accedit.

69. Clausilia deltostoma, Prodr. MS.—Tab. 6. f. 37, 38.

C. testa turrita, gracili, obtusa: anfractibus planiusculis. striis rec-
tiusculis, crebris, elevatis sculptis: apertura oblique obovato-rotundata,
deltoidea, effusa, posticé angustata, sub-biplicata; plica antica columel-
lari, interna, obliqua, duplici; postica simplici, prominente sc. extror-
sum ad marginem peristomatis producta eique continua, sinum ad an-
gulum posticum aperture efficiente: peristomate continuo, expanso,
reflexo, disjuncto.

Long. 5—5} lin. Diam. 14. Anfr. 10—11.

a. anfractibus convexiusculis; sutura distincta.—f. 37. |. c.

Hab. in Madera et Portu 8S”.

B. anfractibus planatis; sub-obsoleta.—f. 38. 1. c.
Hab. in Madera.
Vol. IV. Part I. I

66 Mr. Lowe on the New Plants and Land Mollusca, §c.

Clausilie labiate Sow. (C. bicanaliculate sec. Fer.) nimis affinis;
sed tripld minor; gracilior; anfractibus, etiam in £., minus planis;
sutura mints obsoleta; plica posteriore apertures prominente, margini
labri continua, nec interna; peristomate minis incrassato, nec labroso.
Variantur etiam et a. et 6. collo aperture magis minusve producto.
Clausilia retusa (Bulimus retusus, Oliv.) etiam forma et habitu magis
affinis; sed striolis exilissmis aliarum interstitia decussantibus differt.
70. Clausilia exigua, Prodr. MS.—Tab. 6. f. 39.

C. testa parvula, turrita, gracili, obtusa: anfractibus planiusculis,
omnibus transversé creberrimeé striatis: apertura obliqué obovata, _bi-
plicata; plica antica valde interna; postica prominente, margini peris-
tomatis producta, continua, sinumque cum labro efficiente; ambabus

columellaribus, simplicibus: peristomate continuo, reflexo, posticé sub-
sinuato.

Long. 3—3} lin. Diam. 1. Anfr. 8.
Hab. in Madera.

Clausilie parvule Leach, ut videtur, affinis.

3. Familia, Cyclostomide.
VII. Genus, CYCLOSTOMA, Lam‘.
71. Cyclostoma lucidum, Prodr. MS.—Tab. 6. f. 40.

C. testa globoso-conoidea, nitida, leviuscula, sub-imperforata: an-
fractibus convexis, transversé sub-striatis; sutura impressa.
Axis 2 lin. Diam. 2}. Anfr. 5.

Hab. in Madere humidis sylvaticis.

Testa, quoad formam, Valvatam piscinalem referens, fusca, olivaceo-
cornea, lucida. :

ff.
f.
f.
ff.
f

£12:

2
3
4.
5

OS Di

TABULARUM EXPLICATIO.

Tas. I.

Prant of Goodyera macrophylla, nob. natural size.

A single flower with its germen and bractea.

The uppermost of the three outer petals of the perianth.
Two lowermost of ditto.

The uppermost of the three outer and the two inner petals of the Peri-
anth; the three cohering upwards by means of the former at the back
of the two inner.

The same two inner petals, separated.

Side view of f. 5.

Labellum with column and anther-case, in situ.
Side view of the same.

Same as f. 9, with the anther-case lifted up (artificially) shewing the two
Pollen-masses in situ.

Inside of anther-case, shewing the dissepiment.
Front view of the two Pollen-masses, in situ.

All the figures, except f. 1, more or less magnified.

Tas. IT.

Plant of Tolpis crinita, nob.; smoother and with more entire leaves than
usual.
The more common state of the same.
A seed with its pappus; magnified.
12

12.

a
i ct Sie TS ll

29
S

i
_

Tabularum Explicatio.

Tas. III.

Branch of Sedum fusiforme, nob.

A single petal with its gland, and stamen; magnified.

Tas. IV.

Branch of Ononis dentata, Sol.

Vitrina Lamarckii, nob. \b, state of the same in which the yolutions are

Helix

Tas. V.

a, common state.

visible internally to the apex.
furva, nob, var. a.
erubescens, mob. var. a.
sub-plicata, Sow. [Testa (“ viva” dicta) junior|
undata, nob. ;
phlebophora, nob.
arcta, nob.—magnified.
fausta, nob.—ditto.
arridens, ob.—ditto.
Webbiana, »ob.—Two views of same individual.
Bulveriana, Wood,—ditto, ditto,
tectiformis, Sow.
subtilis, 0b,—magnified.
actinophora, mob.
Porto-Sanctana, Sow. var. a, nob.
»—. — B. nob,
pusilla, 20b.——magnified,
bifrons, nob.
paupercula, ob.—magnified.

a, state in which it is found, coated with soil.

t: le
obtecta, nob: is, the same cleaned.

dealbata, nob. var. a. granulata.

Pho kh rh te bh bh Eh Po bh th rh fe fh rb bh Thoth bh rh ph ph be be be fs

Se GP Ss ie Tos aS aes

Helix

Tabularum Explicatio.

maderensis, Wood.
compar, ob.—magnified.
leptosticta, 20b.—ditto.
lentiginosa, »ob.—ditto.

calva, nob,

Tas. VI,

abjecta, nob.

compacta, nob.

consors, 70b.

depauperata, nob.

lurida, nob.

nitidiuscula, Sow.

punctulata, Sow., var. a. nob,
» ——» — £B.-nob.

lauta, mob.

rotula, mob.

polymorpha, mob. var. a. irrasa.
» — —. B. depressiuscula.
> — —. ¥y- arenicola.
eee SO
» — —. e. calcigena.

> — — ¢. pulvinata,
cheiranthicola, nob. Subvar. 1. zonata,
oxytropis, nob.
echinulata, ob,—magnified,
duplicata, ”ob.—ditto,
turricula, nob,
bicolor, nob.—magnified,
C. tornatellina, ob.—ditto,
C, melampoides, mob.

C. triticea, nob. var. a. biplicata,—magnified,

J
S

2
>

Tabularum Explicatio.

Helix C. triticea, mob. var. 3. edentula—magnified.
. ovuliformis, ob.—ditto.

. gracilis, nob.—ditto.

. lubrica, Mull. var. nob.—ditto.

. anconostoma, nob. var. a.—ditto.

. cheilogona, nob.—ditto.

. sphinctostoma, ob.—ditto.
. monticola, 2ob.—ditto.

. calathiscus, 2ob.—ditto.

. cassida, nob.—ditto.

ol @) @) ©) ©) Tele eke mie!

Clausilia crispa, nob.—ditto.
deltostoma, nob. var. a.—ditto.
—__———, 700. —. (3.—ditto
exigua, nob.—ditto.

Cyclostoma lucidum, ob.—ditto.

Il. On the General Equation of Curves of the Second
Degree.

By AUGUSTUS DE MORGAN,

OF TRINITY COLLEGE, CAMBRIDGE,

AND PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF LONDON.

[Read November 15, 1830.]

Tue object of this Paper is to draw attention to some
properties of Curves of the Second Degree, by means of which
the reduction of their equations from one set of axes to another
is materially facilitated. Little, if any, notice of these properties
has been taken, nor do I remember to have seen their existence
mentioned with the exception of two very limited particular
cases, viz. that the sum of the squares of conjugate diameters,
and the parallelogram formed by them are constant.

Suppose that any curve of the Boeone degree is referred to
axes which make an angle @ or that vy = 0. Suppose the ori-
gin removed to a point whose co-ordinates are m and n in the
directions of x and y, and moreover suppose the directions of
the axes to be changed so that t= 9% sip nae x and y’
being the new co-ordinates; and let hy =~-—o=6. Let the
equations of the curve referred to the first and second systems of
axes be

ay +bxy tex +dy +ex +f =0, (1)

a ay? +bvy +e a "yi +éut+f'=0; (2)

72 Mr. De Morean on the General Equation of Curves

From the suppositions made respecting the axes
wih sin (0—@) x sin (@—W)

pes i
sin @ 3 sin 0 YT
_ sing, sin fa (3)
= tie, Ri

If we substitute in (1) these values of x and y, the resulting
coefticients of y”, xy’, &c. must be proportional to a’, b’, &c. Pre-
serve this condition and make the substitutions and developments

as follows:
Let A =a—hcos 0+€¢ cos*6,
B =sin 0 (b— 2c cos 9),
C =csin’ 8,

4
D=2an+bm4+d—cos 0(2cm+bn+e), (4)

E =sin 0(2cm+bn+e),
F =an'+bmn+cem+dn+em+/f.

In which it is important to observe that if the primitive
co-ordinates be rectangular 4=a, B=b and C=c. If in addition
the origin be not changed D=d, E=e, F=f. Also that if the
axes of x’ is parallel to that of « and wy’ = 90°, the equation
becomes

Ay + Bry + Ca’ + Dy + Ex+F=0.
The substitution above indicated will now give the following

results :

ra! sin? @=A sin? + B sin cos y + € cos’ y,

rH’ sin?@=2 A sin y sin p+ B (sin yy cos p t+cos v sin p) +2 C cos ¥ cos p,

Ac’ sin? @= A sin? p+ Bsin ¢ cos p + C cos , (5)

rd’ sind =D sin +E cos yy,

re sin 9 =Dsin p+ E cos ¢,

Af’ =)
where \ is any quantity whatever.

of the Second Degree. 73

From which we find
A*(b°—4a'c’) sin’ @=(B* — 4 AC) sin® (—) = (b* —4ac) sin® @ sin? 6
b°—4a'c —s B—Aace
2 =— —
Mee sin? @ (6)
of which one particular case is that the parallelograms described
about conjugate diameters are always the same.

Again we find that
ad+c'—b' cos _ a+ce—b cosé

7 (7)

sin’ @’ sin? @

of which when divided by (6) one particular case is that the
sum of the squares of conjugate diameters is always the same.

When

a+e—bcos@=0.

it indicates an equilateral hyperbola. Again, we find that
3cd*t+a'e"—b'd'e' _ cd’+ae*—bde—(b*—4ac) (an°+bmn+cem'+dn+em) )
sin® 6 a sin® @ (
from (8), (6) and the last of (5) we deduce that
ed°+ae?-Ude cd’+ae’—bde
x 1) = tei ay

b?—Aa'e’

(9)

If each side of this be nothing, the equation represents, either
two straight lines which intersect, a point, or is impossible.

If the new co-ordinates be such that 6’=0 we find from the
second of (5)
24 tan ¢ tan y+ B(tan f + tan W)+2C=0.
If in addition to this the new axes must be rectangular, in
which case tan ¢ tan ¥+1=0, we find the following equation

sin 0 $b—2c cos 0}

— : 10
a—b cos 6 +c (cos’ 0—sin* 6) (ao)

from which two values of y are found differing by 90°, either
of which may be the value of ¢, and the other of ¥.
Vol. IV. Part I. K

74 Mr. De Moraan on the General Equation of Curves

The lines determined by these angles are parallel to the
principal diameters. The next question is, how can the greater
principal diameter be distinguished from the lesser? It must
be observed that in the general equation

a _ rectangle of segments of the axis of x _ (diameter parallel to axis of «)’
c rectangle of segments of the axis of y (diameter parallel to axis of y)?”

Since the square of one or both of these diameters may be
negative, that which is numerically the greater can be ascer-
tained only when we know the sign of c*—a’. If this be positive
the greatest of the two diameters above-mentioned is parallel to
the axis of y, &e.

In (10) one of the values of x must be less than 90°. Take
this for the new axis of x, that is, let sin 2¢ be positive. From
(10) it appears that

C-A

5 B
sin 2¢= 7 cos 26 = yy (11)
where M=+.,/B’+(A-—C)*, and since sin 2¢ is supposed positive,
M must be of the same sign as B. Also sin )=cos ¢, cos y= -sin #,
since y—@=90°. Therefore from (5)
ra’ sin’ =A cos* d—B sin } cos P+ C sin’ 4,
re’ sin’ @=A sin’ +B sin p cos P+ C cos’ g,
A(c' +a’) sin? A= A+ C,
d(e’—a’) sin? @=(C— A) cos 264+ B sin 26=M from (11).

Therefore c’—a’ has the same sign as M(4 +C)
BIAS),

since B and M have the same signs; the hypothesis bemg that
the principal diameter which is nearest to the axis of x in the
positive direction is parallel to the axis of x’. Accordingly there-
fore as B(4+C) or sin 6 (b—2c cos@)(a+e—bcos#) is positive or

of the Second Degree. 75

negative, the less or greater principal diameter is nearer to the
axis of x.

When B(4+C)=0 there are two cases to be distinguished.
If 4+C=0 the curve is equilateral, but can only be an hy-
perbola. If B=o0 and C—A is not =o from (11), one of the
principal diameters is parallel to the axis of x. Which it is may
be determined from the sign of c*— a’.

When B=o and C—A=0, or which is the same, b=2c cos
and a=c, the position of the principal diameters is indefinite :
that is, the curve is a circle.

To determine the magnitude of the principal diameters, the
equation must be reduced to the form

ray’ + Ac'x* +f’ =0, where 6 = 90°.

In this case the equations (6), (7) and (9) become

—ANa'c' = Le! 2
sin°@
, , —b
isa _ate cos @

sin’ @

cd’ + ae’— bde
Ni b?—Aac ay

Whence the squares of the principal semidiameters or = ;

and — 7 are contained in the formula

cd’ +ae’*—bde
b?—Aac i

— 2 sin? Qn
a+c—bcos6+,/(a—c)*+b°—2 cos 0(a+c)b—2ac cos)

In this way the curve might be referred to the conjugate
diameters which make a given angle 6’, and the limits of the
value of & might be determined.

K 2

76 Mr. De Moraan on the General Equation of Curves

When the asymptotes are the axes of co-ordinates, the equa-
tion of the hyperbola is \0'2'y7/ + Af’= 0.

The position of the asymptotes is then determined from the
equation
A tan? x + B tany + C=0,
sind b—2ccos@+./b?—4ac
2 ° a—bcosé+ccos*d ”

or tany =

where the two values of x are those of ¢ and y¥.

The equations (6), (7) and (9) become
Nb? _ b —4ae
sin?6’~—s sin? @

rU'cos8’ — a+c—hcosd

sin? 0 sin? 0

,_ ed’ +ae—bde
ae Ne aimedarye

G i bP
whence tan@’= + Bin ake 4 ae

a+c—bcosé
and the equation is
b’—4ac ,,, ed? +ae?—bde
= J(a=0)' +b —2c0s0(ate)b—2aecos0) % * b’—Aac vas
In referring to the expression for the principal semidiameters, it

cd’+ae>—bde
~ b=4ae +7?
the greater diameter is possible, or the curve lies in: the acute
angles of the asymptotes, and the contrary.

appears that if a+c—b cos @ has the same sign as

If the equation of the parabola be reduced to the form
a'y*\+éa' =0, where 0=90°, the equations (7) and (8) take the
following form

ee a+ce—bcos@ a—2./accosd+e
i, sin? @ Th sin? @

of the Second Degree. 77

ed'+ae’—bde _ (/cd—,/ae)’

sin? @ sin’ @

Nae? =

whence the equation becomes

en Jae

sin? @x=0,
¥tqr 2,/accos0 +c)! 3

The position of the axis is determined by the same criterion
as that for finding the major axis of the ellipse; and the curve
may be further ascertained if we recollect that it must lie en-
tirely on one side of the right line whose equation is dy +
ex+f=0, and on that side in which the co-ordinates of a point
are such as to make dy+ex+/ of a different sign from a or ec.

In order to find the co-ordinates of the vertex, we must have
recourse to equations (5) recollecting that B’-—44AC=o0, and that
tang@tany+1=0. From the third and fourth of these we find

ies Aa Ja—,/ecos 0

Tire kil 6: Geno,
But the fourth and fifth of (4) give the following equation
D_ 2,/ak+d—cos0(2,/ck+e)

EE (2,./cek+e)sin@
where k=,/an+./em

and the last of (4) gives
k?+dn+em+f=0,

from the two first of which we find

= (e,/a+d/c) cos 0—(d,/ate,/c)
k= + {+N ee
: Jan sfem= a—2,/accosd+e

dn+em=—f—k’,

and from the last

from which equations m and x may be found. The expressions
present nothing remarkable.

73 Mr. De Morean on the General Equation of Curves, &c.

The consequences of these formule might be carried further,
but what has been here said may be sufficient to turn the
attention of elementary writers to this part of the subject.
Analogous formule may be obtained for Surfaces of the Second
Degree, which will probably form the subject of a future
communication.

AUGUSTUS DE MORGAN.

III. On the Nature of the Light in the Two Rays
produced by the Double Refraction of Quarts.

By G.B. AIRY, M.A.; M.G.S.;

LATE FELLOW OF TRINITY COLLEGE; PLUMIAN PROFESSOR OF ASTRONOMY AND
EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE:

AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read February 21, 1831.]

I propose in this paper to offer some conjectures as to the
nature of the light forming the two rays produced by the double
refraction of quartz; to describe the experiments on which they
are founded ; and to explain the calculations by which the theory
and the experiments are compared. The subject is one to which
(I believe) no attention has been paid, except by one distin-
guished foreigner; the mode of calculation is original to me.
and is, to the best of my knowledge, new.

It is well known that the rays produced by the double re-
fraction of calc spar, (calcareous spar, Iceland spar, or rhom-
bohedral carbonate of lime) and most other doubly refracting
crystals, are entirely polarized: one in the principal plane pass-
ing through the ray (or, if a biaxal crystal, in the plane equally
inclined to the planes passing through the ray and the two
axes) and the other in a plane perpendicular to the former.
From the exact agreement of the phenomena of depolarization
with the calculations made on this hypothesis, we are justified in
supposing that the law holds true when the rays are so little

80 Proressor Airy on the

separated that it is difficult to observe them in the common
mode of inspection. Now it has generally.been supposed that
the two rays of quartz are polarized in the same way: differing
from those of cale spar only in the magnitude and direction of
their separation. It was known however almost as soon as
Arago and Biot commenced their observations, that there is
some anomaly in the rays passing in the direction of the axis
of quartz; and the latter of these observers established the
difference of right-handed and left-handed quartz. Fresnel by
a simple experiment* (which I have repeated) shewed that the
light in the direction of the axis of quartz is not one ray, but
two rays moving in the same direction, and with different ve-
locities. He shewed moreover that a new kind of light may be
produced by causing polarized light to undergo two internal
reflections in a glass rhomb with certain angles, the plane of
polarization bemg inclined to the plane of incidence at an
angle of 45°; and that this light is exactly similar to one or
other of the two rays abovementioned according as the plane
of polarization is on one or the other side of the plane of in-
cidence. And by a mathematical investigation, of which I am
unable to supply the deficient steps, he shewed that the effect
of the internal reflections is to retard by one quarter of an un-
dulation the undulations perpendicular to the plane of incidence,
so that in the light thus modified the particles of ether which
were originally in a straight line will at any time be found
in the form of a circular helix, and each will revolve ‘uniformly
in a circlet. And from the nature of the original experiment

* It is not easy to make this experiment in a satisfactory manner. If the axes of the
crystals are not precisely adjusted, several images will be seen. I have not succeeded in
obtaining two only, though I haye made the others much more faint than the two principal.

+ As I cannot appreciate the mathematical evidence for the nature of circular polarization,
I shall mention the experimental evidence on which I receive it. 1%. The light when re-

ceived

Double Refraction of Quartz. 81

it appeared that in right-handed quartz it is necessary to sup-
pose the right-circular polarization transmitted with the greater
velocity; in left-handed quartz the contrary. I have repeated
and varied most of Fresnel’s experiments relating to this sub-
ject, and am perfectly convinced of the correctness of his views.

Now if, in the experiment with the glass rhomb, the planes
of polarization and incidence be inclined at any other angle
than 45°, the magnitudes of the undulations parallel and _per-
pendicular to the plane of incidence will no longer be equal:
but the alteration of their periods will be the same as before.
The displacement of the particles of ether will still be repre-
sented by a helix, but instead of being traced round a circular
cylinder, it must be supposed traced round an elliptic cylinder.
This modification may properly be called (as Fresnel has called
it) elliptical polarization*. This term has since been used by
Dr. Brewster to express the nature of the light (probably iden-
tical with this, or nearly so) reflected from metallic surfaces.

ceived on an analyzing plate or tourmaline presents the same appearance in whatever direction
the analyzing plate is turned round the incident ray. 2%. The phenomena of depolarization
are the same in whatever direction the analyzing plate is turned. 3". If the polarized light
passes through two such rhombs placed in similar positions, the plane of polarization is
shifted 90°. 4. If they are placed in crossed portions the plane of polarization is unaltered.
5. The phenomena of depolarization agree with the calculations founded on this supposition :
in uniaxal crystals, where the plane of polarization of one ray is changed 360° in going round
the axis, the alternate quadrants are pushed in and thrust out one quarter of a tint: and in
biaxal crystals, where the plane of polarization is changed only 180° in going round the axis,
the alternate semicircles are altered in the same manner.

* If I might venture to fix on the discovery of Fresnel, which among all his wonderful ad-
ditions to optical science appears likely to possess the greatest practical value, I should select
his invention of the mode of producing circularly-polarized or elliptically-polarized light by
internal reflexion of plane-polarized light in glass or water. He has given us the power of
producing light whose laws ure as well known as those of plane-polarized light, and which is
more manageable, inasmuch as it admits of degrees in its ellipticity. The beautiful geometry
of Malus is forgotten when we think of the discovery of polarization: the far more valuable
theoretical discoveries of Fresnel will lose their preeminence when put in competition with an
invention which enables others to make discoveries.

Vol. IV. Part I. 1,

82 Proressor Airy on the

Should any difference be found, I have no hesitation in fixing
on the modification above described as that to which it ought
in propriety to be attached.

I am now able to explain my conjectures on the nature of
the light in the two rays of quartz.

1. I suppose the ordinary ray to consist of light elliptically
polarized, the greater axis of the ellipse bemg perpendicular to
the principal plane; and the extraordinary ray to consist of light
elliptically polarized, the greater axis of the ellipse being in the
principal plane.

2. I suppose that when the ordinary ray is right-elliptically-
polarized, the extraordinary ray is left-elliptically-polarized : and
vice versa.

3. I suppose that the proportions of the axes of the two
ellipses are the same: each proportion being one of equality
when the direction of the ray comcides with the axis, and be-
coming more unequal, according to some unknown law, as the
direction is more inclined to the axis: the minor axes of the
ellipses having sensible magnitudes when the rays are inclined
10° to the axis. '

4. I suppose that the course of the rays after refraction can
be determined by the construction given by Huyghens for calc
spar, with this difference only, that the prolate spheroid for de-
termining the course of the extraordinary ray must not be sup-
posed to touch the sphere for determining the course of the
ordinary ray, but must be entirely contained within it.

These conjectures were originally suggested by the desire of
finding some connecting link between the peculiar double re-
fraction in the axis discovered by Fresnel*, and the double

* It does not appear, I think, that Fresnel had made any distinct supposition as to whether
the two rays in the axis should be considered as the ordinary and extraordinary ray in their

ultimate

Double Refraction of Quartz. 83

refraction commonly recognized. All the phenomena of colours
which I have observed agree perfectly with the results of my
hypotheses.

I may mention that I have found observations corresponding to
many detached parts of the phenomena which I have viewed
assembled, in the early memoirs of Arago and Biot. But the
method which these philosophers used, (particularly the latter),
of examining a small part only at a time, does not appear to
be well adapted to the discovery of the laws of light. In the
experiments which I am about to describe, every thing depends
on the form of the coloured curves; and to attempt to dis-
cover this from observation of detached parts would be perfectly
hopeless. These coincidences I have recognized only since I
made my own observations.

It must be observed that all the phenomena mentioned be-
low are described as they appear when examined with an ana-
lyzmg plate of unsilvered glass. If a plate of tourmaline be
used, the right and left parts of the image will have the same
relative position, but the upper and lower will be interchanged :
the observer’s eye being supposed to turn in such a manner
that the axis of the tourmaline appears wp and down.

PHENOMENA.

I. If a plate of cale spar cut perpendicular to the axis be
examined with the polarizing and analyzing plates crossed, the
system of rings is that represented in fig. 1. If the analyzing
plate be turned less than 90° either way round the incident ray,
the system of rings is that represented in fig.2: and if turned *

ultimate state, or not. From all that I could extract from his Memoirs, I was always in doubt
whether both the ordinary and the extraordinary ray in the neighbourhood of the axis ought
not to be considered as divided each into two circularly polarized rays.

84 Proressor Airy on the

exactly 90°, it is that of fig.3. The order of colours does not
sensibly differ from Newton’s scale, beginning with black.
These are common and well known phenomena.

II. If Fresnel’s rhomb of glass, mounted as represented in
fig. 4, be placed to receive the polarized light, so that the plane
of reflection pass through the divisions 45° and 225°, the calc
spar will present the appearance of fig. 5. The rings are
abruptly and absolutely dislocated: those in the upper right-
hand quadrant and the quadrant opposite to it are pushed from
the center by one-fourth of an interval, and those in the other
quadrants are drawn nearer to the center by the same quantity.
The line separating the quadrants is no-where black: the in-
tensity of its light is uniform and about equal to the mean
intensity. If the plane of incidence pass through 135° and 315°,
the phenomena of adjacent quadrants are exactly interchanged.
No alteration is made by turning the analyzing plate round the
incident ray: the lines dividing the quadrants are always pa-.
rallel and perpendicular to the plane of reflexion at the ana-
lyzing plate*.

Ill. If the plane of reflexion in the rhomb pass through
0° and 180°, or through 90° and 360°, the phenomena are precisely
the same, and undergo the same changes as those in Pheno-
menon I. If while the plates are crossed the rhomb be turned
gradually from the position 0° towards 45°, the rings are gradu-
ally changed, at first becoming (as far as the eye can judge)
elliptical, and then assuming the form represented in fig. 6.

IV. If a plate of quartz, whether right or left-handed, be

* It is proper to mention that I had exhibited this phenomenon to the Cambridge Philo-
sophical Society in the spring of last year, long before the publication of Dr. Brewster's
valuable Memoir in the Phil. Trans. for 1830, and (I believe, but I do not recollect the date)
before its communication to the Royal Society.

Double Refraction of Quartz. 85

interposed between the crossed plates, a set of rings is seen as in
figs 7, 8, 9, 10. As far as the eye can judge, the rings are ex-
actly circular, but there is no black cross, and the central tint
is not black, but removed from it by a number of tints in
Newton’s scale proportional to the thickness of the quartz. Thus
with a thickness 0,48 inch, the central tint is pale pink: with
a thickness 0,38 inch, the central tint is bright yellowish green :
with thickness 0,26 inch, it is a rich red plum-colour: with
thickness 0,17 inch: it is a rich yellow.

The colours then appear to be nearly the same, beginning
from the center, as in Newton’s scale, beginning with the tint
representing this central tint. At a considerable distance from
the center four dark brushes begin to be visible, in the same
directions as the arms of the black cross in calc spar.

V. Now (supposing the crystal right-handed), if the plate
of quartz be thin, and the analyzing plate be turned, the upper
part towards the observer’s left hand, a blueish short-armed
cross appears in the center*, which on turning further becomes
yellow: and the rings are enlarged. On turning still further,
the cross breaks into four dots. The rings are no longer cir-
cular, but of a form intermediate between a circle and a square,
their diagonals (as well as the cross) being inclined to the left
of the parallel and perpendicular to the plane of reflexion. See
fig. 11. If the analyzing plate be turned the other way, there
is no cross: the form of the rings is changed. from circular
nearly as in the former case.

VI. If the plate of quartz be thick, the dilatation of the
rigs and the change of form are all the perceptible phenomena.

* This may be considered as the definition of right-handedness of the crystal: and this
observation gives the readiest means, with a thin plate, of determining whether it is right-
handed or left-handed. If the plate be thick, the easiest method is to observe in which
direction the analyzing plate must be turned to make the rings dilate.

86 Proressor AIRY on the

And on turning the analyzing plate continually to the left, the
rings continually dilate, and new spots start up continually in
the centre, and become rings. If the crystal be left-handed, the
remarks in this and the last article apply equally well, sup-
posing the analyzing plate turned in the opposite direction.

VII. If Fresnel’s rhomb be placed in the position 45°, and
the light thus circularly polarized pass through the quartz; on
applying the analyzing plate, mstead of rmgs there are seen two
spirals mutually inwrapping each other as m fig. 12. If the
rhomb be placed in position 135°, the figure is turned through
a quadrant. If the quartz be left-handed, the spirals are turned
in the opposite direction. The central tint appears to be white.
With the rhomb which I have commonly used (which is of
plate glass, but with the angles given by Fresnel for crown
elass) there is at the center an extremely dilute tint of pink:
I think it likely that this arises from the error in the angles,
as the intensity of the colour bears no proportion to that m
other parts of the spirals. The figure was drawn from the ap-
pearances given by a plate of quartz 0,26 inch thick.

VIII. If two plates of quartz of equal thickness, but cut
one from a right-handed and the other from a_ left-handed
crystal, be attached together, and put between the polarizing
and analyzing plates, the left-handed slice nearest to the polar-
izing plate, the appearance presented is that of fig. 13. Four
spirals (proceeding from a black cross in the center; which is
inclined to the plane of reflexion) cut a series of circles at every
quadrant. The points of intersection are in the plane of re-
flexion, and perpendicular to it. This is the simplest way of
describing the form: but if we followed the colours which gra-
duate most gently, we should say that the form of each is
alternately a spiral and circular are, quadrant after quadrant.

Double Refraction of Quartz. 87

Ata distance trom the center the black brushes are seen. If the
combination be turned so that the right-handed slice is nearest to
the polarizing plate, the spirals are turned in the oppesite direc-
tion. This is one of the most beautiful phenomena of optics.
The slices from whose appearance the figure was drawn are each
0,16 inch thick.

I shall now proceed to explain the mode of calculating these
phenomena on assumed Jaws of the nature of light in the two
rays of crystals.

In fig. 14, let 468, CD, be two parallel rays of the same
pencil incident on a plate of cale spar cut perpendicular to its
axis, of which one furnishes the ordinary ray BH, and the
other the extraordinary ray DE, which afterwards pass in the
same direction EF. (The extraordinary ray of AB, and the
ordinary ray of CD are not to be considered here, as they do
not emerge at H: but each of them will interfere with some
other ray). The paths are found by this construction. Draw
GK a tangent to a circle whose radius is GHx (preserving
Biot’s notation); draw LN a tangent to the ellipse whose semi-
axes are LM xa, LMxb. The velocity and direction of the
ray will be represented by the radius joining the point of in-
cidence with the point of contact, the velocity m air being
represented by GH, LM. Hence the path of the ordinary ray
(measured by the path in air which it would have described in
the same time) exceeds that of the extraordinary by

GH LM

Putting 9 for the angle of incidence, and 7' for the thickness
of the plate, this is found (after all reductions)
= = 1./1—6 sin? 0—./1—a' sin’ 6}.

When 6 is small, this is nearly = 7’ x ee x 6. Call this 0.

88 Proressor Airy on the

I shall suppose with Fresnel, that by a ray polarized in one
plane is meant a ray whose vibrations are entirely perpendi-
cular to that plane; that consequently the vibrations forming
the ordinary ray in the crystal are entirely perpendicular to the
principal plane passing through that ray, and that those form-
ing the extraordinary ray are wholly parallel to that plane.

I*. Now suppose a pencil of polarized light to fall with a
small angle of incidence on a plate of cale spar cut perpendi-
cular to its axis. Let us conceive ourselves looking in the di-
rection of the incident ray; and let fig. 15. represent the pro-
jections (on a plane perpendicular to the incident ray) of the
planes of polarization of the polarizing and analyzing plate, and
of the principal plane of the crystal passing through the inci-
dent ray. Let P:Ap, the plane of polarization (or of reflection)
at the analyzing plate, make an angle « with P,Ap, the original
plane of polarization; and let CAc the principal plane of the
crystal passing through the ray make the angle ¢ with the
former plane. The displacement of the particles of ether pro-
duced by the wave as originally polarized, may be represented

by c.sin an (vt—x), where is the interval of space between two

waves, « the distance measured from any arbitrary point, ¢ the
time since the ether at that point was at rest, and v the velocity of
the wave. (This applies even after the wave has passed through
any media, provided we take for « the space which would
in the same time have been described in air). And this dis-
placement is entirely perpendicular to P,p,. This may be re-

solved into e.sin = .(vt — x).cos (a + ¢) perpendicular to AC, and

* The Phenomena, and the investigations corresponding to them, are numbered in the
same way.

Double Refraction of Quartz. 89

¢.sin = (vt—.x).sin (a+) parallel to 4C. The former of these

furnishes the ordinary ray, the latter the extraordinary. Now
after they have passed through the crystal, we may still keep
the expression for the vibration of the ordinary ray, provided we
make the proper alteration in the value of «: but (by what has
gone before) we must then suppose the path of the extraordinary
ray shorter by ©. Consequently after passing through the cry-
stal, the vibration produced by the ordinary ray is

c.sin <7 (vt—2) -cosa+q@ perpendicular to AC,
and that produced by the extraordinary ray is
c.sin 27 (vt—a+ 8).sina+@ parallel to AC.

When these are received on the analyzing plate, those parts only
are transmitted to the eye or to the screen, which are perpen-
dicular to AP;. They are

a ==
e.sin * (vt 2). cosa +p. cos

and c. sin <™ (vt—# + 0). sin a+@.sin @:

and the sum of these represents the magnitude of the vibration
which comes to the eye, or falls on the screen. It may be put
under this form

sin —* (v¢—2) {e.cos a+ p.cos p+¢.cos " O.sin a+ @.sin 9}

+ cos *2 (vt—«).c.sin =7 ©. sin a+@.sin d.

Vol. IV. Part I. M

90 Proressor Airy on the

Now it must be remarked (as a general theorem which we
shall use hereafter without further explanation) that an ex-
pression of the form

E. sin —* (vt—2) + F. cos =" (vt—2)

may always be put under the form

JEFF. sin = (ve —2£ = :
where tan G=5, and G is constant for that ray. It is plain
that this expresses a periodical vibration similar to that which
we have all along supposed, and whose coefficient instead of ¢ is
./ EF? +F*. It is convenient to take the square of this coefficient
as the measure of the intensity of light: and thus E°+F” will
represent the intensity in all cases similar to that before us.

In the present instance, the intensity or the sum of
the squares of the multipliers of the sine and cosine of
“ (vt — x) is

c feos? a + p.cos’ pd + sin’ a + P.sin’ p
Qa _—— — .
+ 2eos— O.sina + P. cosa + p.sin P. cos pt

= pa + cos 2.a+ $.cos 2h + cos =" 6. sin 2.a + p.sin 2ht.

Thus we have a general expression for the intensity of the
light when polarized light passes in any one direction through
the crystal, and after being reflected by the analyzing plate, 1s
recéived on a screen. If we suppose polarized light to fall in all
possible directions (within certain limits) upon the crystal, we
must give all possible values to @ and ¢, and we shall have the

Double Refraction of Quartz. 91

intensity of light on all the different parts of the screen. It
will be remarked that @ is very nearly proportional to the radius
vector of the corresponding point on the screen, and ¢ the angle
measured from the lower part towards the right.

If the eye be placed to receive the light coming from dif-
ferent parts of the analyzing plate, the appearance will be re-
versed with regard to right and left, and here the angle ¢ must
be measured from the lower point towards the left.

1*. Let the plates be crossed, or a=90°:
cos 2.a+gm=—cos2p: sin2.a+pP=—sin 2g:

and the intensity is

2
<. sin’ 2p. sin® st

or putting for © its value, the intensity is

(ge wp fix @— ,,
3: sin 2p.sin —* a)
This is 0, or there is darkness, if ¢ = 0, or = 90°, or=180°, or = 270°,
whatever be the value of 6. This shews that there is a black cross
through the center, parallel and perpendicular to the plane of reflection.

Also there is darkness whatever be the value of @, if

Tr #0
mr YD

.=0, =7, =2n, &e.,

2br 4br

. 2 a =
or if 0 =0, a (iP ee b?)’ T (a —b*)’

&e.
This shews that there is a dark spot at the center, and a succession
of dark rings, of which the difference between the radii diminishes con-

tinually. Czteris paribus, the squares of the diameters of these rings are

, a—b :
inversely as — 5’ oF are least in the crystal where the double refrac-

92 Proressor Airy on the

tion is greatest; and are inversely as 7’ or are least when the thickness
of the plate is greatest. They are directly as \, and consequently are
greater for red rays than for blue. Hence after a little time the bright
rings of one colour correspond with the dark of another. This gives the
peculiar coloured character to the rings: it also prevents any of them
from being totally black: whereas the evanescence of light in the cross
is independent of d, and the cross is totally black.

2°. Let the plates be parallel or opposite: or a= 0. The expression

for the intensity becomes

ri}

5 §1 +c0s* 2 + 08 2 ©. sin’ 2g}.
If ¢ = 0, or = 90°, or = 180°, or = 270°, this becomes c®: thus there is a
bright cross instead of a dark one. or other values of @ the light is

TT, Ree cas
greatest if aoe = 0, or = 27, &c., and least if Fas =r, or=3n, &e.:

in the former case it =c*, in the latter c’cos*2p. These indications
point out exactly the form of fig. 2. In fact it is easily seen from the.
expressions that the intensities in corresponding parts of fig. 1. and
fig. 2. are precisely complemental.

3". In the general case, if sin 2p=0 (that is if @=0, or=90°, or
=180°, or=270°) the expression becomes

r0| &
20| 9

{1+cos 2.a+.cos 2p} = — $1+cos Qa}.

This indicates a faint cross, which is bright when a is small, and dark
when a is nearly = 90°. And if sin 2.a + = 0, another cross of equal
intensity is found, inclined to the former at an angle a. Generally if

be between 0 and 90°— a, the intensity is greatest when =T@ = 0,

= 2, &c., and least when “2 O=7, =37, &e.: but if @ be between

90°—a and 90°, the intensity is greatest when =" 9 =7, =3n, &., and

Double Refraction of Quartz. 93

least when it=0, =27, &c.: and the same holds if we increase all these
angles by 90°. Thus there is in fig. 3. a mixture of parts of the two

systems of rings in figs 1. and 2.

II. Now let a Fresnel’s rhomb be interposed, and let RAr,
fig. 16. represent the plane perpendicular to the plane of internal
reflection, and making the angle 6 with the plane of original

Jn the ety 2
polarization; the rest as before. The vibration c.sin <_ (vt — 2)

perpendicular to P,A may be resolved into

2
e.sin +" (vt—a).sin B parallel to Ar,

2 2
and e.sin (vt—2x).cosB perpendicular to Ar

Of these the former, by Fresnel’s theory, is retarded one quarter
of an undulation, or the latter is accelerated as much. Adopting
the latter supposition, we must suppose that after the emersion

from the rhomb, the vibrations are

c. sin = (ta). sin B parallel to Ar,

aed, Q
and ¢.sin =~ (vt —2) + 90°. cos B, or ¢. cos ~~ (vt— 2’) .cos

perpendicular to Av. These will furnish for the ordinary ray of
the crystal

2 Nae : = ;
¢-sin -— (vt—x). sin B.sin B+a+pt+c. cos = (vt—a).cos B.cosB+a+
and for the extraordinary ray

2 am =a 9 —————
—¢-sin5~(vt— 2). sin B.cosB +a+ p-+e.cos—~(vt—2). cos B.smB+a+¢.

94 Proressor Airy on the

After emerging from the crystal, we must (as before) diminish 2
in the latter expression by 9: and thus the vibration of the
extraordinary ray

=-—c.sin | (vt—# + @). sin 8. cos Bt+at+o

+¢.c0s =" (vt — x + ©). cos B.sin B+ a +g.

The only parts of these transmitted by the analyzing plate are
the resolved parts perpendicular to its plane of polarization = vi-
bration of ordinary ray x cos @ + that of extraordimary ray x
sin ~

=c.sin = (vt—z).sin B.sin B+a+.cos

Cee oe ee
+ ¢.cos—" (vt — x) .cos B.cosB + a + pcos p
- Qr , —_————- .
—e.sin — (vt—a+ 6). sin B. cosB+a+p.sin p
Qar _ so
aa c.cos =~ (vé—a + @) . cos B.sin B+a+q@. sin ©.
The coefficient of sin = (t— 2) is
c.sin B.sin BHatp. cos p—e.cos—" O.sin B.cos B+atg.sin
Dr. ——— SS
—¢.sin — 6. cos B.sinB+a+p.sin p:
: Ir c
the coefficient of cos a (vt—2x) is
c.c0s B.cosB+a+p.cos p—c.sin—" O.sin B.cosB+atp.sin g

2 _ SS
+c.cos—* @. cos B.sin B+a+q¢.sin .

Double Refraction of Quartz. 95

The intensity, or the sum of the squares of these coefficients, is
(after reduction)

Sii+ cos 23.cos 2p.cos2.B+at+g
Ir 5 F ——— . Aa ; ,
+ cos —— @. cos 23. sin 2p.sin2.B+a+p— sin | 0. sin 28. sin 2p}.

Now if B=45°, cos28=0; sin2B=1;
and the intensity = Sji-sin =o . sin 20.

1". Since a has disappeared from this expression, the figure will be
the same whatever be the value of a, that is, whatever be the position
of the analyzing plate.

2". When ?=0, =90°, = 180°, = 270°, the expression becomes =

which shews that there is a faint cross parallel and perpendicular to the
plane of reflexion at the analyzing plate.

3". When ¢ is >0< 90°, or >180°< 270°, the intensity is greatest

Bae 3 aie 5 4
if Sharia e, &e. and least if 70 = = i soi &. When ¢ is
>90" < 180°, or >270° < 360°, the intensity is least if — Q= = MEE he,
-0 20 a On
d — == —— :
and greatest if = te) Ap ae &e

4". If 8=135°, the expression becomes
c 5 . aa ; 2
Tig + sin > @.sin 2:

from which it is easily seen that the bright parts of the quadrantal rings
in this case correspond to the faint ones when $=45°: and vice versa.

III. If in the last experiment the rhomb be placed in posi-
tion 0, we must make 8=0, which gives for the intensity

e —— ON pe b een
3) + cos 2p.cos2.a+p-+ cos 7 ©: sin 2p.sin2.ar gh,

exactly as in the general case of experiment I.

96 Proressor Airy on the

If the rhomb be in position 90°, the expression is exactly
the same.

If the analyzing and polarising plates be crossed, we must
make a=90°, and the general expression becomes

S {1 -eos 28.cos2p.cos®. B+

— cos" ©. cos 28. sin 2p. sin 2.B+p— sin <7 6. sin 2B. sin 2g}.

2
Here if ¢=0, =90°, =180°, =270°, the intensity is = sin’ 2B,
which shews that there is a faint cross parallel and perpendi-

cular to the plane of reflection. For other values of ¢ the only
variable part is comprehended in the two last terms or

k : seas ‘ ——
— = sin 2p. }sin 28.sin Ot cos 2. sin 2.B +p cos" Of.

This may be put under the form A.cos (= e- B),

where

A=—Ssin 29 sin’ 28 + cos*2.sin*2. B+, and tan B= ee oe

The equation to the dark rings will be found by making
=" 9— B=0, or = Qn, or = 47, &e.;

hence 9 =*8, or =n 428, or = 20 +98, &e.
Tv

and 6= V Tog VE oe V gg VIE,
or = V Fe V renee &e.

Now when £ is small, sin 2.8+¢ being positive, tan 28 is small,

and tan B is small: except sin2.8+¢ is small, when B sud-

Double Refraction of Quartz. 97

denly becomes=90°: when sin 2.8+¢@ changes its sign, B changes
its sign, and is = — 90°; its magnitude then diminishes till sin
2.8+p=—1: and it then goes through the same changes. The
circle is therefore changed into the form represented in fig. 17.
But the rings at the parts where sin 2.8+@=0 being very faint,
the bends of the curve scarcely attract the attention, and the
figure appears elliptical. But when £ increases, the intensity of
the rings where sin 2.8+=0 is not small, (it is represented by
4), and the change of form is easily seen*. All these conclu-
sions correspond perfectly with observation.

IV. To investigate in a similar manner the appearances pre-
sented by plates of quartz, on the suppositions made in the begin-
ning of this paper, we must resolve a plane-polarized} ray into
two elliptically polarized rays. In fig. 18. let AP, be the plane
of primitive polarization: AC the principal plane of the crystal ;
the ordinary ray being elliptically polarized, will consist of one
vibration in the direction Oo,, and another in the direction 0, 0,
following it one quarter of an interval of undulations: the coefti-
cient of the latter vibration being=s x that of the former, where
k is a fraction depending by some unknown law on the incli-
nation to the axis, but becoming=1 when the inclination=0, and
=0 when the inclination is considerable. And the extraordinary
ray will consist of one vibration in the direction Ee,, and another
in ¢,¢, preceding it one quarter of an interval: the coefticient of

* I have investigated (in nearly the same manner) the form of the curves, supposing
the crystal placed between the polarizing plate and the rhomb. The calculated pheno-
mena are nearly the same as those described, and agree perfectly with observations.

+ I use this term instead of rectilinearly-polarized, the natural derivative from Fresnel’s
substantive, only because it is shorter.

Vol. LV. Part I. N

98 Proressor Arry on the

the latter being = i x that of the former. Let the vibration per-
pendicular to AP, be c.sin = (vt—x) or c.sin— (for brevity): and
let the vibrations be
In Oo,, p.sinE+v, or pcos v.sinE+p sin v.cos & or w sin £ +2 cos &.
In 0,0,, kp.sin §+v—90°,
or —kp.cos€+v, or kp sin v.sin §—kp cos v.cos &, or £x sin E—kweosé.
In Ee,, q.sin F+yx, or g cos x.sin + g.sin x. cosé, or y.sin E +x. cos &.
In ee, 2. sin F+x%+90°,

Fi : : q iis Y
or 1 cos é+ x or — 4 sin y.sin E +r 7, C08 x - C08 E, or — 7, Sin + 700s &.

Resolving these in directions parallel and perpendicular to AP,,

and comparing them with the vibration from the original polari-
zation,

cos a+ (w sin E+& cos t)+sin a+¢@ (kw sin &—kw cos £)
+cosat+¢@ (ysinE+s cos£) + sina+@ (— z sine + cos €) =e sin &,

sin a+@ (w sin E+ cos t)—cosa+@ (ka sin &—kw cos é)
+sin a+ (y sin E+8 cos £) — cos ato (— ; sin — + cos £) =0.

Or (since these equations ought to hold for all values of é)
equating separately the coefficients of sin € and cos é,

i aie = e= —— sin
&f TEP wth sin oF G.2 +e oF G-y — Ge He coevee (1),

cos atp.a—k snat+gd.w+cosatp.® + sete y=0

Double Refraction of Quartz. 99

sinat+$.w—k.cosatp.xtsinatg.y + S0*P, 9 noBtes (3),
sinat+$.2+h.cosatg.w+sinatg.x —SSFPy_9 aproce (4).

By the solution of these equations

Cc —_—
w = Tap esate,

oe é c.sina+
Toe Peery,

Y= Tye ©: 008 at+@,

| aero
s=— +p °: sin at @.
And hence the expressions for the vibrations are (omitting the

c
common factor ee
Se ear -_ —— Qa
In 00, cosa+@.sin ZL (et 2) +h. sin at+¢. cos TL (et- 2).

In 0,0,, #. sin a+q@.sin =" (vt—x)—k.cos a+. cos = (vt—2).

—_

In Ee,, I. cos a+ p.sin = (vt—x)—k.sin a+.cos =" (ota).

In e,¢., sina+q@, sin <7 (vt—2)+h.cos a+. cos = (vt—z),

After emerging from the crystal, « must be diminished, in the
two latter expressions, by 9. (0 is in fact negative for quartz,
but that circumstance makes no difference in our investigations
or conclusions),

N 2

100 Proressor Airy on the

Now when the light is received on the analyzing plate, the
only parts sensible are those perpendicular to its plane of re-

flexion. They are, (putting — as before for <7 (wt —2)),

From Oo, cosa+.cosp.sinE+k.sina+.cos.cosé.
From 0,0,, #.sina+@.sing@.sinE—k.cosat+¢p.sing.cosé.

From £e,, #. cosat¢. cos. sing + “70 —k. sina+¢. cos. cos E+ 220

ae +k.cosa+.sind.cos’+ "=

From ee, sina+g.sing.sing +

Taking the sum, the coefficient of sin — is

Ia i ha ali

“7 Inr09
+ k.sina.sin ea es: sin @. -COS——.

The coefficient of cos & is

ksina +k’. cosat+.cosp.sin =70 isin a .COS a0) sina +.sing,. sin =
The sum of the squares of the coefficients is (after all re-
ductions)

ae 8 Sanaa ce ae 2
(1-2)? cos'a+ 2g. sin? — if (i iad + 2k. sin a.sin™*) :

And restoring the multiplier a5 we have for the brightness

qd 7

sin a. sin — 7).

eGaen cos’.a + 2o.sin— + c° (co neers
17) Oe p. 7 ( 8a. CO8 >— + Ti

Double Refraction of Quartz. 101

When the plates are crossed, or a = 90°, this becomes

1 ay aA ocr \S) ; Ak? ie TO)
2 (9) - = ace = ep =
c G aie -sin’ 2. sin aia +e a+ FP) sin’ >
aE a0) i WS es
= (is ~ —_ ¢ —_______ —— 9g ,
ee aN te + hk’) Ca TOTES SI cay of

1. For any value of @ this is 0 when

a
= =0, =7, =2m, &e.

This shews that there are dark rings, exactly circular: it represents cor-
rectly the experimental fact.

2. But since by our 4" hypothesis the spheroid and the sphere, used
for determining the course of the two rays, do not touch, © will have
some value when @ is 0, It cannot therefore be expressed simply by

Tx x 6,

but must have an additional term 7'x H. The value of E (as depending
on A) may be thus found. At the center & is supposed (hypothesis 3) to
=1. Consequently the general expression for the intensity of light at the

center is

A 79 5 . wO\* ; Fs 7rO
c (cos «..cos = ate sin asin” ~) or c’.cos (« — =):

This is 0 when a — “e = 90°: or putting 0’, a’, for these particular values
of © and a, a =90 + = Now it was found by M. Biot that in a
right-handed crystal, a’ (measured in the direction that we have supposed)
must exceed 90° by a quantity proportional to the thickness of the plate
directly and the square of \ inversely. That is,

ut LL

eT
ne one Nerd

I hig os

102 Proressor Airy on the

whe e a-b
EF. therefore is 3 and consequently 6 = 7'x (— + eRe 0 y:

We may remind the reader (as we shall have occasion to use it afterwards)

that eh is the angle through which the analyzing plate must be turned

from the crossed position to produce darkness at the center for the particu-
lar colour used. We may also remark that, if we still consider © as posi-
tive (which we shall continue to do), all our expressions (as appears from
this comparison of theory and observation) must be understood to apply to
a right-handed crystal: if the sign of & be changed, they will apply to a
left-handed crystal. But if, more correctly, we put a negative symbol for
©, then our expressions would apply to a left-handed crystal, unless the
sign of & were changed.*

Oo. Te Tr(@-B6),.
The value of + 8 therefore et a Ge

that \’ + 0A may represent the length for rays of any colour, \’ being that

Let us suppose

for one of the mean rays, and therefore constant, and dA being small for

: ; 9
all the bright colours. Then for ara aa? We have

Te , Tr(ad - 38’) 9 orate | Tr(a’ — 6) e

x2 en) ar xy a ay aa ) nearly.
The mixture of colours in the rings will depend only on the difference of

the values of 7° for different colours, and not on its absolute values, and

* The reader who will take the trouble of tracing the expressions will find that, if the
sign of © and of & be changed at the same time, not only will the right-handed-ness of the
erystal remain the same, on comparing the expression with Biot’s experiment, but also all
the directions of the spirals &c. in the succeeding experiments will remain the same. Thus
the connection between the right-handed-ness and the direction of the spirals is independent
of the sign assumed for ©. With this consideration, I have thought it best to use the same
symbol in the theorems for cale spar and for quartz.

Double Refraction of Quartz. 103

consequently only on the last term of this expression.* Now this is the

same that we should have had for the colours in Newton’s scale depending

: 2T T x(a? — b°
on the thickness Vv en ae 7 ) 6 of a plate of air, if

a—b

ab be the

2 72
same for all colours: or if the variations of — be proportional to the

variations of ), the last term must be altered in a certain proportion.

and our statement will still be true. It appears therefore that the central

tint will be nearly that corresponding in Newton’s scale to a plate of air
whose thickness is fa : and will be followed by the other orders of New-

ton’s scale. This agrees sufficiently with the observations.

8 : 3
3. Unless =A = 0, or =7, &e. the light is not = 0: therefore there

is no black cross. But the light is least when ¢=0, or = 90°, or =180°.
or = 270°: and greatest when @ = 45°, or = 135°, or = 225°, or = 315°.
This shews that there are dark brushes parallel and perpendicular to the
plane of reflexion, but not interrupting the rings. As & is nearly = 1 in
the neighbourhood of the axis (by hypothesis) they are not sensible near
the center: but as, on removing to a distance, k approaches to 1, they

become stronger. This is conformable to observation.

* This is not strictly true: for though, in comparing this with one of Newton’s rings, the
colours mixed may be the same, their intensities will not be equal except sin? = be the
same, and consequently the compound colour will not be the same. But it is plain that, if

we take the ring preceding and that followiug the point where the difference of ae is the

} 2 : 9 :
same as in the case before us, then at the points where sin? — has the same value for the

mean rays, we shall have mixtures differing in opposite ways from that under consideration.
In the same manner, in the rest of the paragraph, it must be understood that, if we take
those colours of Newton’s scale between which the colours of these rings lie, we shall always
advance in the scale: but we may possibly have no ring intermediate between two of New-
ton’s, or we may possibly have more than one.

104 Proressor Arry on the

V. Taking the general expression for the brightness, and

: Qk
making tan y = i+ F tan a, we have
Praia) + as sina. sin = V cos?a + ae sin* a. cos es
A Seal kr TI a. — = on ‘ palais
paige Oo hot x SST a haa pag aa aid

and the general expression for the brightness is

RE RECN DN ee. 8 iO Bins
© (Fp) - 00s" a+ 2G. sin* = +e (cos! a + Gap sina) . cos! = —

Supposing that is not much altered by a small alteration of 0,
this is a maximum or minimum for a given value of ¢ if

—— . 2r9 S12 : f 3
0= (1-2). costa + 2p .sin ~~ (I+ Ff. cos’ a + 4 sin®a). sin27? oy,

279
or tan

= tan a x Lt Pe costa + ah sin’a + 1 — # ° costa + 2h
1+ 1+#} .cos’a + 4H sin?'a —1 — kh}. cos?a + 2h

Therefore an8 will be greater than ¥ (or Y +7, or ¥ +27, or

+37, &c. for each of these satisties the equation tan yaaa tan a)

by the angle » whose tangent is the second side of the expression.

1. If & be nearly equal to 1 (which we suppose to be true when @
is small) and a less than 90°, the expression for tan w is always positive :

its greatest value, when

¢ =90°— =, or 180°— >, or a1 = or 360° —'<-

2
" 9, A =o 212 ene
is Bang ae 1 Brine,
1+ 4k? —1 — #|?. sin®a

: gone on De a!
and its least, when p = 45 3° 8 135 ? OF 225-5, or 315° — 5,
2k

1S l+F tan a.

Double Refraction of Quartz. 105

279
r

+7, or Y + 27, &. by an angle w which is always included between 0

Therefore in the bright or dark rings

will be greater than y, or

and 90°, and which has its maximum at a part which, when a is less than
90°, is found by looking a little to the negative side of the perpendicular
and parallel to the plane of reflexion (thus, if the crystal be right handed,
we must look to the left of the upper part, and the points of the square
appearance will be found in that place). When a is greater than 90°, the
maximum value of the tangent takes place for points found by looking a
little to the positive side of the perpendicular and parallel: but the tangent
is then negative (for tan a which enters as a multiplier is negative). Con-
sequently the maximum contraction of the circle is found by looking to the
positive side, and the points of the squares will be found by looking to the
negative side. Whichever therefore be the direction in which the analyzing
plate is turned, the circles will be changed into the form represented in
fig. 15 (the crystal being supposed right handed). This remarkable conclu-
sion agrees perfectly with the facts of observation.

2. If % is very small, the expression for tan w may become negative,
which shews that w will suddenly exceed 90°, and after having continued
so during the change of @ through an are of various extent (according to
the value of a) will suddenly become less than 90°. This shews that the
form of the bright and dark rings will be that of fig. 2, except that in-
stead of absolute interruption of the rings by the eight radii, they will pass
very highly inclined through those radii. The rings of quartz become so
faint at a distance from the center that I have not been able to observe

whether this is or is not supported by fact.

3. We have already noticed that when a = 90° + ce there is a dark

spot at the center. Now for any given value of 0, it appears (from the
Vol. IV. Part I. O

106 Proressor Airy on the

general expression) that the light is least when

p = 45° — + 135° — =, &e.

This shews that, with this value of a, the spot will be a darkish cross, its

arms in positions

a ote
45 — 3? 135 3° &e.

But these are exactly the angles at which the depression of the ring below
a circle is a minimum, or at which the points of the square are found.
Therefore, we may expect to find a cross-like spot in the center, its arms
in the diagonals of the square. This corresponds perfectly with the phe-

nomena.

4. The succession of colours in the cross-like spot, it is easily seen,
depends only on this circumstance: that as A is greater for red than for
blue rays, the value of a which allows no red rays to pass is less than that
which allows no blue rays to pass. That is, for a certain small value of in
there are rays of the blue and only of the spectrum transmitted: for a

larger value, the rays only of the red end are transmitted.

5. If the polarizing and analyzing plates are parallel, or a= 0, the

expression for the brightness becomes

2 2 +e @ sho 72
ce -— 31 tee - COS 2p} sin a °

It is easily seen that this indicates a series of rings, but there is now no
total darkness. When

¢=0, =90°, = 180°, = 270°,
the expression is

5 6 4h? 79
C 2

eae xo

Double Refraction of Quartz. 107

As & becomes smaller, this varies less with the variations of 0, but does
not vanish: that is, in receding from the center, the rings are more and
more interrupted by a white cross. If the plate of quartz be thin, it may
happen that the first ring is so large as to be sensibly interrupted. In
this case (as the first ring is broad) it will lose the appearance of four in-
terrupted quadrants and become four dots. This is easily seen in experi-
ment.

VI. We have seen that, for a bright or dark ring, 278 will

differ from ¥~ only by » When a is 0, » is 0: and when a is
90°, » is 90°, having increased gradually to that value. Also when
a is 0, ¥ is O or z, or 27, &c.: and as a increases gradually to 90°,
y increases gradually to

Tv
Se oi Ob cas &e.

27r0 4 F :
Consequently, = or ¥ + increases by 180°, while a increases

from 0 to 90°. In the same manner, as « increases gradually from
90° to 180°, » increases gradually from 90° to 180°, and y from

7 3a 5a
g° or og? OF ae &e. to 7, or 27, or 37, &e.:

2790 Toe
and consequently — or ¥ + again increases gradually by 180°.

The same holds for every successive quadrant of revolution. Thus,
if we fix our attention on any ring, and turn the analyzing plate
to the left (the crystal being right-handed) the ring’s distance
from the center (or 6, which

‘ / 2b Te
TG m\°-=))

will increase continually, but not uniformly. This is conformable

to fact.
02

108 Proressor Airy on the

VII. When Fresnel’s rhomb is interposed in position 45°, we
must suppose AR, fig. 18, to be perpendicular to the plane of
internal reflection, and we shall have the vibrations

= ae oe (ve —x) parallel to AR, and Fi es = Ate — 2)

wee to AR. Resolving these in directions Be and
perpendicular to 4C, we find for the former

= Fa f sin2” (ota). cosa P+ 45+ Fe - COS = (vt—2).sinat pt 45
or — =F sin =" (ot — 2) — (a + 9) — 45°:
and for the latter,
iF sin 27 (vt—2).sin opti Te cos = (vt—2) .cosatp+45",
or 7 cos 27 (wt — x) — (a+ ¢) — 45°.

Now taking the same expressions as before, for the vibrations
in Oo,, &c. and comparing the sums in directions parallel and

perpendicular to AC, (putting 27 (wt —x) — 45°=£), we have
“sin F—( +) or wie E+
ee ——s —\a as! a
Fa Ja a+. sin sy sina +o. cos &
s\.
= (ke - ) sin — + (3 — kw) cos &

c Cia c —_—
Wik a ~—(a+) or ie ee geen er? cme

= (w + y) sin & + (x + 8) cosé

Double Refraction of Quartz. 109

Comparing the coefficients of sin ~ and of cos £,

bs) Cc ———

kz — i x ag ae

thw = Fy nat @
OQ %

wry = ge LP

tb —_——

ee RB ere

Obtaining the values of w, x, y, x, and substituting in the expres-
sions for the vibrations, we find (omitting the common factor

aam a)
(1+ 4h) /9/°
In Oo, (1-4). cosE- (a+).
In 0,0, &£(1—h).sin E-(a+¢).
In Ee, £(1+8) cos —(a +).
In ee, —(1+h).sin F—(a+ 9).
After emerging from the crystal, ¢ in the two latter expressions

2 =
must be increased by ——.

Taking the neath parts of the vibration perpendicular to
AP., we find for the efficient vibration,

(1—-). cos E—(a+@) . cos p+k(1—k).sin €—(a+ @). sin p

+h (1+h).cos&—(a+¢) + 270 cos p—(1+h).sinE—(a+¢) + 270 sin op.
The coefficient of sin —(a+¢) is

k(1—k). sin p—k(1+h).cos @. sin” To (142). sin Pp. cos

hay
A

110 Proressor Airy on the

The coefficient of cos —(a+¢) is

(1—£).cosp@+ (1+4).cos p. cos “20 — (1+4).sin ¢. gin
The sum of the squares is

1+F] -2k.1—#. cos 26+ 2h.1—F*. cos 2p. yee

—(1-').sin 2g. sin =

Restoring the factor ae niet we have for the brightness
cf 2k.(1—-#)
ia il

2k (1—k*) 270
Mey ues 2p + +P) cos 2. cos —

_ p22

. . 2rO
eure ae x \.

; +k?
If we make tan x = aE
E) ok
14+ (1+

tan 29, the two last terms, o1

Ems
cos 2. cos = — sin 2@.s1n sae

a $
a cos’ 2p+sin* 2p. cos pe es
and the expression for the brightness is

c 1-k 4k 2h

= iI ~The va cos’ 2p-+ sin* Wh -+ [ze Cos *#)

aS 1-f Ake
1+F (1+4*)°

become LEE
1+

cos’ 2p+sin’ 2H. cos? ~ 2 +%
1;

As the multiplier of cos*—— +% i

9 is never = 0 while # has any
value between 0 and 1, the rings are not interrupted in any part of their
circumference.

Double Refraction of Quartz. 111

2”. This multiplier however is small when & is nearly=1. It is
also small when £# is small, if sin 2=0, that is, when P=0, =90",
=180°, =270°.. Hence there will be no rings very near the center: and
at a distance from the center, they will be faint in the lines parallel and
perpendicular to the plane of reflexion.

3". The form of the dark rings will be determined by making

cos? 2° + X=0: and that of the bright ones by making it=1. The
first of these suppositions gives

ae) 9e ar lar é iss) a 9e

eg igs Sag Se andi tg 9? Ne.

Now x increases from 0 to 90°, from 90° to 180°, &c., while 2 increases
from 0 to 90°, from 90° to 180°, &e.: consequently x never differs much
from 2: and therefore for the dark rings

tO & 30
ne = 9 _ Pp; ie — Oo _ Pp, &e. nearly.

That is © increases continually as @ diminishes: and consequently @ in-
creases continually as @ diminishes. This shews that the curve is a spiral,
and that (reckoning from the central fold) it is turned in a negative di-
rection; the eye when fixed on a part above the center must turn to the
left to trace the curve as it recedes from the center, supposing the crystal
right-handed. If the crystal be left-handed, the sign of # must be changed :
this changes the sign of x, and the spiral is turned in the opposite di-
rection. This agrees perfectly with observation.

4. If we take the radius vector in the direction opposite to that cor-
responding to any point of the spiral, that is, if we increase p by 180",
we find for the new values of © in the dark rings
30

= ————
5 ~ +180, — P4180, — G+ 180, &e,,

via 30

TT f .
or sah iets eae rm’ &e. nearly.

112 Proressor Airy on the

This is exactly the same series as that for the original direction of the
radius vector: and therefore the values of @ are exactly the same for
any radius vector as for that opposite. But the curve (as we have seen
above) is spiral. These two conditions require that the form of the dark
line be two similar spirals mutually inwrapping each other, their positions
differing by 180°, But no other alteration of @ will give the same values
of © in the dark rings. Consequently the form of the dark rings is two
spirals, and only two, turning in the same direction, and in opposite

positions. This remarkable conclusion is supported by fact.

5". When 2¢ is between 0 and 90°, yx is greater than 2: when
2 is between 90 and 180°, yx is less than 2p: and so for successive
quadrants. That is, when @ is between 0 and 45°, y is too great, or
8 too small, for a spiral of uniform approach to the center: when ¢@ is
between 45° and 90°, © is too great. This shews that the spiral will have
a square appearance, the right-hand angles being higher than the center.
This is precisely the form really presented to the eye.

6". The expression for the brightness of the center (where #=1) is

j é Bie atis z
simply a: As this is independent of \, it shews that there is the same

mixture of colours at the center as in the light which we use: and

that therefore with common light the center is white.

I should only take up the reader’s time unnecessarily by
going through the investigation with the rhomb in position 139°.

VIII. We have found (in the investigation of IV.) that put-
° Qn ‘ P . as
ting & for <_ (vt-2), the expressions for the vibrations when light,
at first plane-polarized, has passed through a plate of rght-

handed quartz, are (omitting the common multiplier i+ a)

Double Refraction of Quartz. 113
In O00,, cosat@.sin E+h.sin a+. cos &.
In 0,0,, sina+q.sin €—k.cosa+@. cos E.

- 270 La)
In Ee,, cosa+¢. sin E 4 =79 _ sin ato. cos — + Ney:

._— . —— 270
In e,e., sin at p.sin E+ =7O 4 heos a+. cos E + ——.

We have now to resolve these into the elliptical vibrations in
a plate of left-handed quartz of the same thickness, with its
principal plane in the same position. It will be observed, that
the difference of right-handed and left-handed quartz consists only
in the difference of the sign of 4.

Taking first the vibrations in Oo, and 0,0,, and using those
letters to express the vibrations in those directions,

Oo,=vibration in O’o', + that in E’e’
=w sin +x cos &+y sin +2 cos &,

0,0,=vibration in o',o', + that in ee’,
=—hkax sin E+hw cos — + ; sin & — 4 cos &.

Comparing the coefticients of sin — and cos é,

a= 1-F
cos at+p=wrty, w= 1 Dy aes at¢,
ksina+p=2 +x, ALS at,
1+
Hsina+o=—hka +=,¢ Whence es,
a Y= 14 pcos at,
—k csatp= hw—¥ a
: ke ® sin atd¢.
~1+k

Vol. IV. Part I. P

114 Proressor Airy on the

Whence the vibrations, after emerging from the second plate,
are (omitting the common multiplier tae addition to the
former) :

In O',0; 1—#.cosa+.sin & + k(1—#’).sin a + p. cos &.

In oo, —H(-#).sinat p.sin & + k(1—F). cosa + p. cos &.

= . ea 9
In Ee’, 2k. cosa +. sin +920 + 2h’. sin a + p.cos— + are.
‘ee a 270 ——, 2rO
nrere. 2H. sin a+ p-sin E + —— — 2k cosa + .cosE + 27°
Similarly for the vibrations in Le, and @,e,:
; 270 270
He, = wsint +——+ x cost + ——
r r
d 270 27rO
+.y sin &.4+.—— + 8003 £ +———-
r r ,
_ », 200 ~ 20
é,e@. =—hkx sin & + ane + kw cosé + a2
Ss 270 y 279
Fe AU ere gee es
: ages : 2790
Comparing the coefficients of sin — + =n8 and cosé + a
@ ee ee — 2
K cosatp=wty, w= pet,
—k si = ae
sts Epes t= sin a+,
i a=. * \ whence
pe oe Teta | tae _-FO-*) way9
Fae op tet Be :
k cosat+p= kw — 5, 2
hk x a6.

' 1+h

Double Refraction of Quartz. 115
Whence the vibrations, after emerging from the second plate,
ae 1
are (omitting TB):
270

In Oo, 2h. cosa+.sin F + =
—2h.sina+@.cos pis 250
In o',0'. 2k. sina+@.sin E+ =e
+2 cosat+@.cos p44 278
In E’e’, —2(1—F*) cos a+¢.sin =
. +k(1—#*) sin a+. cos — + —_.
In ee. (1—4*). sin a+q@.sin — + ———
+k(1—#) cosa+g.cosé + si,
Adding together the vibrations in the same direction, we find

Ai c
(omitting the factor ae aoe

Vibration parallel to Qo,
=1—#'.cosa+o.sinE +h. (1 - kh) sina + p.cos &

SSL OTT er ees 2rO
+ MMP. cos.a+ p.sin E+ 7° ~ 2k (1 ~ #) ‘sina + p. cos E + —

—# (1-H) cos + p.sinE +929 +k(1-2). sina +p .cosé +979,

116 Proressor Airy on the

Vibration parallel to 0,0,=

—k#(1—#).sina+¢. sinE + k(1—#).cosa+ p.cosé

+48 sina tp. sin€ +222 — 9h(1— 2) coset G- cost + =T—

4 (1—#). sina + p-sin E+ 222 + RF) cosa p. cosk + 2°.

To avoid unnecessary generalities we will suppose the plates
crossed, or a= 90°: which gives
Vibrations parallel to Oo, =
—(1—#). sing. sinE + k(1—#). cos p . cosE
— 4 sing.siné + 2x0 — 2k(1—F) cos p.cosé + =n

+ (1 — ) sing. sing + S224 2-H) cos cos + “T~.

Vibrations parallel to 0,0, =
—k(1— #)cosp.siné — k(1 — #’). sin p .cosé

2
+ 48 cos .sinE + °=" + Qk — If) sin g cos E +22

+ (1-H) cos p. sine + ar0-k(1 — i) sin p. cosE + a8,

The efficient vibration, or that perpendicular to 4P., will be
found by multiplying the former of these by cos ¢, the latter
by sin ¢, and taking their sum. Thus we have

1-#

2 sin 2p .sinE + k(1 — #) cos 2p. cosé

Double Refraction of Quartz.
— 2k(1— k)cos2.cos& + are

+— sin 2p. sing + 22° 4 £(1—#) cos 29. cos + ==—
The coefficient of sin & is
LS 2 F :
- sn 2g + 2k(1 — #) cos 2h. sin ais
4
- : 4
ie Hsin 2p cos @™° — (1 — 2) cos 29. sin kaos
The coefficient of cos ~ is
k(1 — #) eos 2 — 2k(1 — F) c08 2. cos ===

4790

ne + k(1 — k’) cos 2. cos

le" Jis oe :
+ a : sin 2@.sin

The sum of their squares is, after all reductions,
1 PF sin? 2. {4k. cos ag. sin? — 2.(1 + #°).sin 2p. cos@ Ol.

Ga :
———.., we have for the brightness

And restoring the factor cea
L—F\2 rm Ol AE a aks, “ 7
2 a —— —_ = . . ral
ct. (Fs) - sin x SE eos 2. sin x 2.sin 2. cos
If we make tan x= 1+ tan 2¢, this expression bec
we make tan x= —57— tan 2, this expression becomes

ile
e. TR (16H cos* 2p + 4.1 + Af. sin’ 2¢). sin’ 7° sin’ (

This vanishes, or there are black lines, when sin? — = 0, or when

(3)
J ==Q; or =7, or =27, &c.

118 Proressor Airy on the

This indicates a series of dark circles, whose diameters are the same as

those of the circles seen with either of the plates singly.

2. The expression also vanishes when

or when

[=X =rtxy =Ar+yX &e.

Now when 2¢@ increases from 0 to 90°, from 90° to 180°, &c. x also in-
creases from 0 to 90°, from 90° to 180°, &c.: consequently y will never
differ much from 2. So that the expression vanishes when

Toe 29, =7+2¢, =2r+2¢, &c. nearly.

In the curve defined by this equation, it is plain that © increases con-
tinually as @ increases, and consequently @ increases continually as @ in-
creases. The curve therefore is a spiral in such a position that if we look
at a point above the center, and follow the curve towards the right-hand,
the radius vector continually increases.

3. Now if we increase @ by 90°, or 180°, or 270°, we get the same

values for x, increased by x, or 27, or 3. Consequently the values of
the radius vector, at a point of the dark line, are found by making =
equal
in the first of these to m+y, 2r+y, 3rt+y, &e.
in the second to 2a +x, 8rty, 4r+yx, &e.
in the third to 3r+yx, 4r+yx, 5a+y, &e.
These are evidently the same series of values as that found with the origi-

nal value of @. That is, if we draw four radii vectores of equal length at

angles of 90°, and if one of these terminate in a point of the dark curve,

Double Refraction of Quartz. 119

the others will also terminate in points of the dark curve. This condition
can be reconciled with the spiral form of the curve, only by supposing that
the curve consists of four similar spirals, each of which is turned 90° from
the position of that adjacent to it. The general form of the curves will
be therefore four spirals, in positions differing by 90°, and all turned the
same way, intersecting a series of circles. This very remarkable form is

precisely the form given by observation.
4. The intersection of the spirals and circles is found by making both
the equations
sin® ze = 0 and sin? (72 = x) = 0,
to hold at the same time. This gives
xX=90, or =7, or =27, &e:

and consequently
3a

Tv
gp =0, or = 9° or =7, or = 2°
That is, the intersections of the spirals and circles will all lie in the lines
through the center parallel and perpendicular to the plane of reflexion.

This is exactly true in the experiment.
: : : s _ 79 . :
5. And since the successive circles require values of Suecessively in-
creasing by 7, and the successive points of the spirals on the same_ radius
- xO : : : ?
vector also require values of “y ~‘Successively increasing by =, every circle

will be intersected by the spirals in the lines above mentioned, and there-
fore every circle will be intersected at every quadrant. This is verified by
experiment.

6. If we consider the parts near the center and not in the circumference

of one of the circles, the angle ¢’ corresponding to a dark point will be nearly

7 0’ 7 0 é
Qn” or @, 790; &e.

120 Proressor Airy on the

where 9’ is the value of © corresponding to @ = 0,

pare
or O= ath
Consequently,
eT eT A
p FESS eee F &
co) x? F oe + 90 c

But Ss is the angle through which the analyzing plate must be turned
to the left to see the dark spot with the right-handed plate alone. Con-

sequently the dark cross in which the spirals originate is inclined, (the
upper part to the right), by an angle half as great as the angle through
which the analyzing plate must be turned (the upper part to the left)
to see the dark spot with the right-handed plate only. This appears to

agree with experiment.

7. In the whole of this we have supposed the right-handed plate
to be nearest to the polarizing plate. If the combination be turned, with
the left-handed plate nearest to the polarizing plate, we must change the
sign of &. This changes the sign of y (that of @ being supposed the
same); and it will very easily be seen that in consequence of this change,
the direction and position of the spirals will be exactly inverted. This is
found to be true.

8. When & is small, the expression for the brightness becomes small
if sin’ 2@=0, that is, if ¢=0, or = 90°, &c. This accounts for the dark
brushes seen parallel and perpendicular to the plane of reflexion at a great
distance from the center.

9. When ¢ is > 0 < 45°, x is > 2p: when @ is > 45° <. 90°, x is
<2. Observing that the spirals intersect the circles when p=0, =90", &c.,
it is easily seen from this that they cut them at an angle rather
greater than the angle at which a uniform spiral would cut them. This

seems to be observable in the experiment.

Double Refraction of Quartz. 121

10. The peculiar vividness of the colours appears to be explained
by this consideration. The dark curves are defined by making the expres-
sion for the brightness to vanish totally: whereas in some other cases (as
in investigation VII.) the dark curves are defined by making the expres-
sion for the brightness a minimum. Here then the colours are much less
diluted with undestroyed white light than in investigation VII. And in
the lines parallel and perpendicular to the plane of reflexion, y=0, and

: . ee eye oes) ’
the expression for the brightness is ¢ (14m 16H. sin Here then

is no sensible light for a considerable distance nearer to and farther from
; 5 iS)
the center than the point corresponding to = na: the colours are less

mixed, and are therefore more vivid than perhaps in any other phenomenon
of polarization. And the greatest brightness in these lines, if & be not
very small, is four times as great as the greatest brightness in the lines
making angles of 45° with them (as will be seen on making @=45" in
the expression for the brightness). Experimentally the colours are brightest

in the lines parallel and perpendicular to the plane of reflexion.

It is almost unnecessary to point out to the reader that none
of the peculiarities of these appearances would exist if & were
either 0 or 1, that is, if the light were either plane-polarized or
circularly-polarized. None of the expressions however would be

altered, if, instead of & we put i: that is, it is indifferent which

ray we suppose to have the major axis of its ellipse parallel to
the principal plane.
From the agreement between the observed and the calculated
appearances, I think there is little doubt that the nature of the
Vol. IV. Part I. Q

122 Proressor Airy on the

light in the two rays of quartz is such as I have described.
I do not mean to exclude the possibility of supposing that the
form of neither wave (in the construction for determining the
course of the rays) is exactly spherical or exactly spheroidal:
provided the difference of the forms be nearly the same as that
of a sphere and a spheroid. Nor do I mean to assert that each
elliptically-polarized ray consists exactly of two plane-polarized
rays following each other at the mterval of one-fourth of an un-
dulation: or that the ratio of the two axes in the two rays is
exactly the same. But I conceive it to be perfectly certain that
the general character of the light is such as is stated in my hy-
potheses.

I have not made any calculations upon other suppositions,
but I can hardly imagine that any other would represent the
phenomena to such extreme accuracy. I am not so much struck
with the accounting for the continued dilatation of circles, and
the general representation of the form of spirals, as with the
explanation of the minute deviations from symmetry, as when
circles become almost square, and crosses are inclined to the
plane of polarization. And I believe that any one who shall
follow my investigations and imitate my experiments, will be
surprized at their perfect agreement.

There is one relation between the construction for determin-
ing the course of the rays, and the nature of the rays, which
deserves (I think) particular attention. It is that (comparing
the rays of quartz with those of any other crystal) a change
in the nature of the ray is accompanied with an interruption of
continuity. The nappes of the wave surfaces are absolutely se-
parated. This is not the case in the common construction for
uniaxal crystals, nor in Fresnel’s construction for biaxal crystals.

Double Refraction of Quartz. 123

There may possibly be a connection of the same kind as that
between the change from partial reflexion to total reflexion
within glass, and the accompanying change from plane polarized
light to elliptically-polarized light. The cases are at least thus
far analogous, that the change in the light and the interruption
of continuity go together. But we are so much in the dark re-
specting the physical constitution of quartz, that we cannot at
present go farther.

It might have been desirable to verify my suppositions by
more direct experiments on the separate rays of quartz. I can
only plead that the duties of my office have not allowed me
the necessary time. They would (under all circumstances) have
been much more troublesome than those which I have the honor
of laying before the Society: and I do not think that they
would have been more satisfactory. The appearances presented
by depolarization are admirably adapted to the discovery of the
most delicate differences in the nature and course of rays. The
same want of time I hope will be allowed as an excuse for the
want of accurate measures*: without which no theory, however

satisfactory in general explanations, can be considered as firmly
established.

G.B. AIRY.

OxpseRVATORY, CAMBRIDGE,
Dec. 30, 1830.

Soc Lo ee a ee ee a eee eee

* It is much to be wished that the rings and spirals exhibited by quartz may be
accurately measured, their diameters in different directions ascertained, and a comparison
with theory instituted, by means of Biot’s measures of the doubly refractive energy of quartz.
Should the observations and the theory disagree, it would shew, either that there is
some latent error in the theory, or that the difference of curvature of the sphere and
spheroid near their vertices is not the same as that which is inferred from Huyghen’s
construction, modified as above

Q 2

aye

vila > had sane edie eo GV
Botenitor tei) os anstnaliony

Yarviwwieeg ntilayeon itty

eiiwt> hor ani etait lta

“at +.

IV. On the Resolution of Algebraccal Equations.

By R. MURPHY, B.A.

FELLOW OF CAIUS COLLEGE; AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.
[Read March 7, 1831.]

INTRODUCTION.

Tue researches of Lagrange on that part of Pure Analysis,
which forms the subject of the present Memoir, have been fol-
lowed up with considerable success by many foreign Mathe-
maticians, amongst whom M. Augustin Cauchy deserves to
be particularly distinguished ; indeed the extensive use of the
theorem of Lagrange in Physical Astronomy, had turned the
attention of Analysts to consider more intimately the nature of
that series, the conditions of its convergence, the root which it
particularly represents, &c. I have referred to as many papers
on this subject, scattered through the Memoirs of the French
Institute, the Journal of the Polytechnic School, and the Annales
de Mathematiques, &c. as I conveniently could. I do not find
that in the point of view in which this subject is here exhibited,
I have been anticipated in any of the articles above referred
to, a point on which it is necessary to be doubtful, without
actual reference, from the great number of persons who have
been recently, and are at present, engaged in extending the limits
of Analysis.

126 Mr. Murpuy on the

I have supposed the given equation to be such as to contain
neither negative nor fractional powers of the unknown quantity 2 ;
if such should enter any proposed equation as ¢(«)=0, we need
only put «=a+s, and consider = as the unknown quantity, since
@(a+z) may always be expanded according to the positive and
integer powers of x, when no particular value is assigned to 4a,
this is therefore to be understood, unless where the contrary is
expressed.

Suppose the root of an equation ¢(x)=0 is sought, the fol-
lowing simple rule which I have proved and applied in Section 1.
will give it with great facility.

“ Divide the given equation by 2, take the Nap. log. of the
quotient by means of the formula

take the coefficient of the first negative power of x in this loga-
rithmic expansion: this, with its sign changed, is the root of the
proposed equation.”

If the proposed equation were of x dimensions, it has x roots,
and it is natural to enquire which root is given by the preceding
method. I have in the same Section shewn that it analytically
gives a result which comprehends all the roots, but that arith-
metically it gives the least root.

If, instead of the root of an equation, any function /(x) of the
root should be required, there is given in Section (2) for this
purpose, a rule nearly as simple as the above; namely,

‘Take the same Naperian log. as before. Multiply it by
the derived function ,f’(z), the coefficient of the first negative

Resolution of Algebraical Equations. 127

power of x, with its sign changed, will be the required function
of the root; minus the same function of 0.”

In Section 3, there is given a method equally simple with
the former ones, to obtain the sum of any specified number of
the roots; also the sum of any given function of a specified
number of the roots.

In this Section it is also shewn how to find the m' least root
of an equation, or any function of it, to which are annexed
some remarks on that relation of imaginary quantities, which cor-
responds to the relation of greater and less in real quantities.

As the sum of m roots exceeds that of m-—1 by one root,
it is clear that by this method we can get all the roots of the
equation, as well as any function of any root.

The principles laid down in the first three Sections are ap-
plied in the fourth to the deduction of several theorems of ana-
lysis; the theorems of Laplace and Lagrange are simple and
almost immediate consequences: a theorem somewhat similar to
Lagrange’s, which M. Cauchy gives in (Vol. IX. Memoirs of the
Institute,) for the sum of any function of all the roots of an
equation, I have here shewn holds true for any specified number
of the roots as for the whole, though the series only terminates
in the latter case, and under particular conditions.

There is also a theorem given by Burmann, which is of great
use in transforming series, but the ordinary demonstration is very
long, and may be seen in the notes at the end of Vol. III.
Lacroix, Diff. Calc. but in the present method it follows in a
few lines.

When we revert a proposed series, or find the root of an
algebraic equation in a series, the law of the latter is not

128 Mr. Murpuy on the

readily visible: the same Section contains the expression of this
law.

There is besides in this Section, a remarkable expression for
the correction to be applied to the root of an equation, when to
the equation itself there is added a small term.

Next in order, the important subject of the Limits of La-
grange’s series is considered.

Section V. treats on Definite Integrals: in this Section the root
of any proposed equation, any function of the root, &c., are all
represented by means of Definite Integrals; and we here see the
Analytical use of the multiplicity of the values of certain tran-
scendent functions, when the quantity under the sign of such a
function should be the same at both the limits of an integral.

The last Section contains various points of Analysis, connected
with the present subject, which could not conveniently be brought
imto any of the others.

Resolution of Algebraical Equations. 129

SECTION I.

To find the root of any equation ¢(«)=0, which contains only
positive and integer powers of «.

Divide the equation by x, take the Nap. log. of the quotient ;
the coefficient of the first negative power of x, with its sign
changed, is the root of the equation.

For example, in the quadratic equation

a’ +ax+b=0,
divide by z, and take the log. of the quotient,
i.e. 1. (a+0+%) ,orla+l. (ogists)

by the preceding rule the root should be therefore the coefticient

1h
of - in
x

and selecting the coefficient of : which enters only in the 1",

3°, 5", &e. terms, we get for the required root

2,848 6.5 8.7.68 a)
a eo 8 eg 8a Ra eS

as it evidently is, since this series represents the expansion of

- 6).

Vol. IV. Part I. R

130 Mr. Mureny on the

This instance of the application of the present method is
sufficient to shew its nature, but as we may frequently facilitate
the operation by a slight previous transformation of the proposed

equation, I have added a few more examples.
To find x in the equation
az’ +%s—6=0,
in this case, the rule may be directly applied; but the result
may be obtained, rather more simply, by putting s-—b=..
The equation becomes
a(a+b)"+2=0,
therefore by the rule

x= coeflicient of * in —1. {1 + “(a +b)"}

* 11 a a a a =
= coeflicient of = in — ~, (« +5) + 4.9 (wt by" — 4. . (a +b)" + Be.

3n.3n — y
OED yay So

2
= —ab" an be Bi
cs 2 1.2

As another example, suppose a+z=1. (x) to find x,
or s—e’.«*=0.
Hence by the rule

Ate 1 es z
= coeflicient of = in —1. (1-«.5)

Resolution of Algebraical Equations. 131

When the given equation contains fractional or negative powers
of z, put (as was before observed) « = = +a, and considering * as
the unknown quantity, we may directly apply the method.

The rule given in this section may be thus proved. Suppose
@ (x) to be resolved into its simple factors; i. e.

(a) =C.(@—a). (w@—) .(«-y)

then £2) =c(1-$) (1-4) (1-2)...

where C’= C.(—£).(—y). &t.
LPM 1c (1-$) +. (1-8) +1 (0-2) + &e

of which the only term which contains negative powers of x, is

p (a)

the coeflicient of : therefore in |. eae is —a, where a is a root of
the equation ¢ (x) = 0.

Supposing the equation of ~ dimensions, which of its roots is
given by this method? The result analytically comprehends all
the roots, but arithmetically it gives only the least root. Let us
recall, for example, the quadratic equation «°+aa2+b=0; and,
supposing «, 8, to be the roots; —a=a+ , b=af; and substituting
these values in the expression for the root given at the beginning
of this Section, we get

aB | @B 4 dB 6.5 aif
o+B (@+Bf. 2 @+py sear 3° (a+B)
RQ

root = ——,, + &e.

132 Mr. Murpny on the

and expanding each term of this series according to the descend-
ing powers of 8, the value of the root is

4 a 5.6 @&
+3 = {1-5 B* 1.2 men Be
6.5 a‘ a
8.76 ,a
tog 4° pu &e
&e. &e.

All the terms here mutually strike out with the exception of the
first a; but since a and # are similarly involved in the given
series, it follows that if we expanded according to the descending
powers of a, all the terms would strike out except 8; thus this
series analytically represents a or ( indifferently; but if we stop

at the »' term in the former case, the error is of the order i) :

B
and in the latter, of (f) ; and if a <8, the former is very small,

and the latter great; the series therefore, arithmetically, gives the
least root (abstracting from its sign); and generally, whatever is
the proposed equation, the series expressing the root is manifestly
a symmetrical function of all the roots, and therefore does not
analytically express one more than another, but comprehends
alike all the roots; but as each term may be expanded in a con-

Resolution of Algebraical Equations. 133

verging form when we make use of the ascending powers of the
least root, the given series arithmetically designates the least root.
And thus it is that Lagrange’s series which, as it will be hereafter
seen, coincides with that obtained by the present method, repre-
sents arithmetically the least root of the proposed equation.*

SECTION II.

To find any function} /(«) of the root of a given equation
p(w) =0.

From {(0) subtract the coefficient of - in Pen ae
f(x) being the derived function or differential coefficient of f (x).
the remainder will be the required function.

After the examples given in the former Section, many will
not be necessary, in illustration of the present principle: we
shall take but one, viz. to find the value of 2 when «=c..«',

n being positive.
The required value of x", by the above principle, is the coeth-

s Te .
cient of - Im — 227-11. (1 — <.'); and expanding the log. and ob-

* When the first power of x does not enter the given equation, put r=a-+z, so as to
introduce a first power of z; and then, let z be treated as the unknown quantity.
+ f (2) is supposed to contain only positive and integer powers of x; should it be other-
wise, we have only to put c=a +2; the same is to be observed of ¢ (a).

134 Mr. Murpuy on the

serving that the first —1 terms may be rejected as not con-
taining any negative powers of z, we get

nm ett ct |

: Ls c
a” =n x coeflicient of ; im }

nx n+1.x
that is,
nm.n+2 2.n + 3”
m — an onthe ert? ort 4 &e,
ar=c"+n 1 ete 1.2.3 Cone + We,

This principle evidently includes that given in the former section
as a particular case, and admits of a proof nearly as simple, viz.

Put ¢(«) in the same form C.4—a.a-fP.x-y¥ &e.

= pia) . , ass = ah £
and .. l. pis LCS ee Oe: Fe ee

the part of which containing negative powers of < is

1 2 3
ee

Hence = is evidently the coefticient of 5 im — vue ;

: us
and .. a" = coefficient of pl ee mee 2),

Let f (ce) be any function of a, as f (0) + 4a+ Be’ + Ca’ &&
putting for 7 1, 2, 3 &c. successively, it follows that

J (a) =f (0) + coefficient of in —-{4+2B2r+3C2* &e.} sf. ei)

= f(0) — coefficient of F in f(x) .1. we

Resolution of Algebraical Equations. 135

SECTION III.

To find the sum of any specified number (m) of the roots of
an equation.

Divide the given equation by x", take the Nap. log. of the
quotient; the coefficient of 4 , with its sign changed, will be the
required sum of m roots.

Ex. (1). 2° +ax+6=0, to find the sum of two roots, take the
coefficient of Lint (1 Aap =) viz. — a, which is evidently the

x x x
sum of the two roots.

Ex. (2). 2°—aa"—b=0, to find the sum of two roots. By
the above rule, that sum

38 a
= coefficient of - in — 1. ei — 5
x x

ax” + _)

ax’+b i. (ax" + by t, (a x” + by?
x? oa 32°

Sad 1
= coeflicient of 7 + &e.

Hence, when x is even, this quantity is nothing which agrees
with truth, since the roots of the proposed equation are then of

the form + ./a.

But when ~ is odd, the terms of this series which involve
1 +1
z are at m places distance from each other, and begin at the “>

term, its value therefore is

Ae 4 3n=1.80-3 @& a’ Wight 5n—1.5n—3.5n—5.5n—7 a’ pp"F ge

—$—$— $< $—$—$<—$—$<———————— ———__———"

Cw ay OM TAGS

136 Mr. Murpnuy on the

When the term involving 2” is wanting in the proposed equa-
tion, then, instead of dividing immediately by 2”, put x= +a,
and the difficulty of having no term free from the unknown
quantity in the quotient (on which depends the expansion of
the log.) will be avoided.

To prove the rule given in this Secticn, suppose a, a2..-an,
@,4,++-a, to be the 2 roots of the equation ¢ (x)=0.

Hence ) (x)=C. (w—a,). (@—ay)......(@—a,,) (@ — ay 43) voeeee (w—a,)5

a ea (1-%)....(0-%)

x (1- z ys} ok (1-2),

An+1

and taking the log. of both sides, it is manifest that

(2)

: es
a,+a,+...4,,=coeflicient of ie if

This method then will give us very simply the sum of
any proposed number (m) of the roots of the given equation ;
but since there may be various combinations of m roots made
from the 2 roots of the given equation, which combination or
group of m roots is that given by the present rule? The answer
is analytically, it gives any possible group of the m roots, but
arithmetically the m least: that it analytically gives any is obvious,
since the resulting expression is manifestly a symmetrical func-
tion of all the roots, and therefore cannot analytically represent
any one combination of m roots more than any other. That it
gives the sum of the m least roots may be thus shewn.

Suppose, first, that the sum of two roots of the equation
$(x)=0 is obtained by the present method, and that a,+ a, is the
particular combination which it gives.

Resolution of Algebraical Equations. 137

p (2)
oe

: ee
Hence a,+«a,=coefticient of 2 Wise 1.

Now since a, is a root, therefore ¢ (x) is of the form x -a,.P;

e iL er: IZ
therefore a, +«a,=coefticient of i ie = ae

: ie
=a,— coefficient of - in I ae

but the coefticient of - in me is the least root of the equation

P=0 (by Section 1).

Hence a, must be the least of the quantities: a, a;, a,, &c.: by
similar proof «, must be the least of a,, a,, a,, &e.; and therefore
a, and a, are the two least of a,, a., a;, &c., that is, they are the
two least roots.

In the.same manner if by the present method we get the sum
of three roots a,+a,+a,, then putting ¢(«) =2—a,.%—a,.P’, we
have 3

-1 8 =-1(1-*)-1(1-2)-12,

x
and taking the coefticients of * at both sides by the theorems

already established, we have
a, +a,+a,=a,+a,+least root of the equation P’=0;
therefore a, is the least of the quantities a,, a,, a;, &e.
“LOTTE, LEGA aaa un DS 1 MN eT 2, G4, as, We.
Aiden audvesins chs as caetaesancedasnconvss @1, Ay, a5, &C.

therefore a,, a,, a,, are the 3 least roots of the equation: thus it
appears in general that this method gives the sum of the m least
roots.

Vol. IV. Part I. Ss

138 Mr. Murpnuy on the

In comparing thus the magnitude of the roots, we go on the

hypothesis of their reality. When the roots are imaginary how-

ever, a similar order may be supposed to subsist ; thus— «+ Wass ——b
s a less numerical root than —a—‘\/ = — b (abstracting from the

sign) when 4 6, it holds therefore the same rank when +; <b.

Since the whole theery of imaginary quantities results from
an extension of the properties of real quantities, we have this
advantage resulting from the theorem above proved, that it esta-
blishes an order when quantities are some real and some ima-
ginary—analogous to He relation of greater and less amongst real
quantities.

To find the m™ least root of a proposed equation ¢ (2) = 0.

From the coefficient of = - inl. ote) subtract the coefficient of } ~

am 1
= )

in 1. , the remainder will be the m"™ least root. This theorem

is an esac consequence of what has been already proved in
this Section.

To find the sum of any function /(a,) /(a.) &. of the m least
roots.

From m/(0) subtract the coefficient of * = im of (a). pec )

a ?
the remainder will be the required Fee

The proof of this theorem is similar to that given for a function
of one root, it will be unnecessary therefore to add it.

To find the value of any function of the m least root.

Resolution of Algebraical Equations. 139

Find, by the last theorem, the sum of the values of that func-
tion for the m least roots, and also for the (m—1) least roots, and
take the difference.

SECTION IV.

In this Section we shall make a few applications of the present
method, to shew with what facility it gives various theorems of
Analysis.

Lagrange’s Theorem.

Let s=a+h F(z) to find f(s) any function of x; put s=2+ a4,
the equation becomes x= F(a +2); to find /(«+ a), apply the
rule in Section (2).

Hence °

hFa+x
fe

F(z) or f(a +a) =f (a) — coefficient of + in f'(a+2) ie

=f (a) + coeflicient of in

h 2 fi ad be See oe ;
7S (a+ 0). Fase) Ces. eee We.

Now if we consider f(a+x) F(a+2), f'(a+«)F(a+2), &c. as so
many separate functions of @+., and expand them by Taylor’s
Theorem, it is visible that the above equation becomes

S (x)= fla) +h f' (a). F(a) + i ee poe &e.

140 Mr. Murpny on the

Laplace's Theorem.

Let x= F§a+ho(z)} to find f(2).

Put s=F(a+u) the equation becomes w=hd.F'(a+u), to find
SF (a+u), or f(z) apply the rule in Section (2).

Hence

f(s) =fF (a) — coefficient of + in fF(a+u)l. 14h AES

and expanding the log. as in the last case, it evidently gives

SO) = fF (a) + hf F(a).p Fla) + 25 (fF la). oF ay + &e.

Burmann’s Theorem consists in this, that if x and uw are two
functions of « which vanish together, and X any other function,

then f. ;
ee de (a) 4
du" asa

putting x and w = 0 after the differentiations.

ts
In fact, if we form the equation «= X"*', the value of a is

’

1
coeflicient of : in -1(1 es -)

therefore coefficient of h’+! in u

' ly: A dX i
= coefficient of - m (+1) .a* = i ae ei
when wu is put = 0 by Maclaurin.
Put the same equation under the form
Bhs
s=h.-.X¥™,
Uu
then by Section (2)
1
axe hi es

phe y
u or AX" = coefficient of 4 in - 2 L(l1--.- 2X");
3 dx BU

Resolution of Algebraical Equations. i41

*. coefficient of A"*' in u

n 1
er Be ea pa ann
= coefficient of = in — . : é (-)
% gr n dx u
pn - 1 OX ax”
= coeflicient of z"-' in ——— ,-—< , (¢)
m.n+1 4 u

nisdX (z%\")
moe,
1.2...n+1dx"-*"

Equate this with the former value and we get the above theorem.

To find the sum of any function of the m least roots of the

equation
(3 — a)" =h. F(z)
as f(a,) aia (cre) etna ST (an):
Put
S=x+a; -. a@=hF (a + 2)
to find

J (a + By) + f(a + Br) + &e.
1, Bx, &e. being the m least values of x, we have, by Section 3, the
required sum

= m f(a) — coefticient of 4 in f(a +2) 1§1— L F(a+x)t:

whence, if we expand the log. this becomes

ae f(a). Fiat ht de ae (a). F(a) }
= h. : a
BT (a) +h 1.2...m—1.da™-} at 2°1.9...9m—1.da"— a

In this remarkable theorem if we put m=1 it becomes Lagrange’s;
but if we put m= the dimensions of the equation, then the m
least are in reality all the roots, so that this case corresponds to
the theorem given by M. Cauchy in Vol. 1x. Memoirs of. the
Institute.

142 Mr. Murpny on the
Let us retake for a moment the same equation
(zs — a)” = h F'(z),

if we extract the m™ root of both sides and suppose 1, p, p’---p”~'

to be the m" roots of unity, we may form m different equations,
viz.

1
St + ped” ae (e)*

to each of these equations apply Lagrange’s Theorem to determine
f(z), we thus get m values, viz. :

S@)=f (0) +0. f a). Fay +5 Af (@) Fary'+ be.

SF (s) =f (@) + phng’ (a). F(a" = eg {f (a) Fy +&e.

1 1 o.m—1 a 30 Ae
SF (8) =f (a) + p"-. hf (a). F(a)” + = if’ (a) F(a)"}' + &e.
Adding up all these equations, and recollecting by the proper-
ties of the roots of unity that their sum, or the sum of any of
their powers is 0, except in the case of the m", 2m", &c. powers.

when the sum is m, we evidently get for the sum of the function

Resolution of Algebraical Equations. 143

of the m roots precisely the same result as before obtained in
p. 141.

We see likewise from this, that if there is an equation of x
dimensions, and we transpose the highest power, so as to put
the equation under the form «x = %/P, then applying Lagrange’s
Theorem, to find x, and multiplying the first group of x terms by

Qn ear, A — .4r ,
1, co Nem cos = + /—Isin—, Xe.

and the 2”, 3", &c. groups of x terms respectively by the same,
we shall get a 2" root of the equation, and multiplying them by

4a — . Ar
118 cost nf =I a ae &e.

we shall get a 3" root, and so on for all the roots.

Suppose a series is to be reverted, or an equation solved, such
as a,=«—a,2°—a,«°, &., we can easily express the law of the
general term in the value of «, thus by Section I.

ae | Pe a A
«x = coeflicient of = in—l. Sy — (% + '4,0+ 4,2", we.)
x ? x 4
A 3 ey lie
therefore the general term of «=coeflicient of > in
X (* + A,X 4+ A,X", &e.) 3
NNV2X

(“ +4,%+a,2°, &e.\" ‘Vy
=coeflicient of a” in € * 67%. 67%", Xe.

Now

1.2...”

Hence the general term of «=coeflicient of 1.2...x—1 i in the fol-

lowing product, viz.

ad,

aa,

(1 + ast + — &e.) (1+aa,0° + &e.),

144 Mr. Murpny on the

, poy LS mal. aya &e.
1. €. W=27 ob, x 1.20 b es

subject to the two conditions
b,+6,+ 6, &e=n,
b,—6,-—2b., &e.=1.
When to an equation F'(x)=0 we add a very small term e.¢9(),
what is the corresponding correction to be applied to the root?
The root of F(x) =0,

is the eoeticient of : in -— 1. erie)

and the root of F'(x)+e¢(x)=0
is the coefticient of 2 ened (F(a) + Cpr) ;
3 x l « x §
therefore the increment of the root

=edamaentiof ano (= “s a) cay Fa
x xv x

ep (x) )
F(z) \

= coeflicient of s in 1. 1 +

eg («)
F (x)

= coefficient of 5 in
when e¢ (x) is supposed to be very small.

Cor. When the proposed equation is s=a+/(*), which by
adding an additional smal] term, becomes

x=at f(s) +eh (®)

the correction to be applied to the root is

e ih (a) + [p (a) fa] + (oa) fer’ + &e.{.

Resolution of Algebraical Equations. 145

To find the value of the error committed when we stop at
the x“ term of Lagrange’s series.

If we pursue the same method as that used at the beginning
of this Section, we get the required error

hn?

= coefficient of = in J (a+2).F(a+a)*?

m+1.art}

Arre
Sareea. (a+2).Fa+x"*", &e.

a : i \ (4. F(a+2)\"*?
= coeflicient of ~ in f (a+2) (=) ra

piesa ae of

Lv “n+2

sate 1. : FE (a--a)\"*} -
=coeflicient of Z in f (ata). f )(—<**) ch

+ (Fy et bat

the integral commencing from h=0.

But the part under the sign of integration may be summed,
and
(eae ig hn
— o ‘

= a ACES)
x

_ F'(a+a)"*! h h
7, a ‘e@—h F(a+2)’

therefore the error = coefficient of «”~" in

he
f (ata). Faray (ae,

to be calculated by Definite Integrals, in the next Section.
Vol. IV. Part I. Ab

146 Mr. Murpruy on the

SECTION V.

Ler F(x) be any function of x, containing only integer
powers of 2, positive, or negative; as

F(«)=A + Br + Ca* + &e.

b
ape ae & + &e.
x x
Put < for x, multiply then by ¢’ and integrate with respect to 9,
we thus obtain

Be? Ce
9 ct ae + &c.

[ F'(e). = const. + Ae? +
“6 :
+b.0—ce-°— &c.

All the terms in this result, except 6. and the const., are cir-
culating or periodical terms; i.e. if we give @ the series of ima-
ginary values comprised in the formula @+«a,/—1, then for any
two values of a, which differ by (27) a whole circumference,
these terms have precisely the same value, but the term 5.0 be-
comes 27b,/—1 between the above limits: thus it appears that
the coefficient of the first negative power of x in

F(z) = Ss MF). €:

the integral being taken through one entire circulation, 1.e. from
9 to 0+ 27f—1.

Restore now for ¢ its value x; then /,F'(e°).e? becomes /, F(x)
which by actual integration is

Resolution of Algebraical Equations. 147
fconst. + 4.2 + B.= + &e.

+b.1. (x) — “ — &e.}.

And the limiting values of «2, which correspond to the former
limits of @ are e’, and &*”?*=?, both of which are equal. But the
integral does not vanish, though the numerical values of the
limits are equal; for it has been above shewn to be equal to

2xb./—1, which may be easily explained ; thus,
Ax or Ae’ between limits= 4e'*?*”-*— Ae’
= Ae’. (e7"-1_-1) =0.

eat! Br alt
Similarly —* between limits=0, &c.

but 61. 2 between limits=d 1. &*?*”~'_—51., é

=b (04+22,/—1)—be
=2rb./—,

though, therefore, the quantity under the transcendent sign of
log. is the same, at both the limits of the integral; yet, if we
suppose x to circulate through a series of values, until it again
arrives at the value from which it set out, we must then make
use of the multiplicity of the values of the log. (x), (which are all
included under the form 2m ./—1 + real log.) by giving to it.
at its limit, that value it acquires by one circulation, namely
Qn ./ —1+real log. (x).

It is the same, with other transcendent functions arising from
integration, and possessing the character of having an infinity of
values in arithmetical progression; as sin-? (0), tan~' (2), &e.:

148 Mr. Murpny on the

the common difference of that progression is the quantity by
which they alter through one circulation ; which is therefore

2a for sin-1 (x); =m for tan-'x, and 27,./—1 for log (x):

if then at the limits of an integral, the quantity under any of
these signs should be the same; and if x has been supposed to
have passed through one circulation, from its leaving a certain
limit, until it has returned to the same, then for the correspond-
ing part of the definite integral, we evidently must not put 0,
but the common difference of the above-named progression.

Thus, in finding the area of a circle - oe taken from

a"
x=1 until «=1 after one circulation, oS .. is sin~? (1) —
sin-*(1), but the former must be manifestly understood to exceed
the latter by 27.

If this distinction be attended to, there will arise no difficulty,
in calculating the values of terms of this nature when they enter
definite integrals, as they frequently do; we have also here the
advantage of seeing the analytical use of the multiplicity of the
values of these kind of functions.

It has been proved, (p. 146), that the coefficient of the first
negative power of 2 in F(z) is equal to {, F(c*).<°, taken from 6 to
6+27,./—1 or through one circulation. Applying this to the
theorems given in the former Sections, we get the following
results.

If p(«) =0 be any equation containing only positive and in-
teger powers of x, then

2r./—1x the least root =— f°. 1. {p(e).c-%

2r./—1x the sum of m least roots =—f, c’ 1. {¢ (e*) .e7 "8.

Resolution of Algebraical Equations. 149

Similarly, 27,/—1.~ any function f(a) of the least root is

1. Feo
= o y IL Sp (e%) .6-%

=20/—1f(0) -
and 27,/—1x the sum of that function of the m least roots is

2mm Jaf (0)— fH. 1. fo ().om

the integrals, in all these cases, being taken from @ to 0+27./—1,
or through one circulation of «. If the functions f(x) and op (2)
contain negative or fractional powers of 2, we must put x«=a+z,
and proceed with x as the unknown quantity.
In like manner the error made by stopping at the x term of
Lagrange’s series given in p. 145.
_ peta’ Fara
ne — het F(a 4 8)
the integral with respect to h commencing when h =
With one application we shall terminate this Section.
Let 2°+ax+b=0, to find z.
Qr./—1x root = — fel. (ce +a + be-)

eF_b

Be ee Oe Noe ti) a pee

Now 4.-¢° evidently vanishes between limits;

e? — 6.68
. 24r,./—-1x root = [—g-—4—

o€ +ae+b

Bag aée*+ 2b
4 I hae th

_ gia’ tae 5), d. 8

2/4, +aerb + * Ciena (Ae

150 Mr. Murpny on the

or 2rJ/—1 . root

tan~ 1 —_——— + constant,

W/o Sine Mb = 4

and taking these transcendent functions (which enter directly by
integration) from @ to 6+27./—1, the expressions under them do
not alter at these limits, and therefore, by the preceding remarks,

=-$1 (6° + aei + db) +

the value of
the log. between limits is 27,/-—1

ot HRD irc date orci socbeta sete s cmpacts TS
cab
. 2a 1, root =-20J-1.5 +5.
=

It need scarcely be added, that this instance is only brought as
an illustration of the operation.

The method, here adopted, for finding the coefficient of the
first negative power of 2, in any function of x; on which the
value of the reot of an equation has been made to depend, was
chosen, as leading to the preceding remarks on Definite Inte-
erals: the application of either Parseval’s Method, or of a theorem
eiven in my former paper, on Definite Integrals ; (Vid. Cambridge
Transactions, Vol. 111. p. 437) to the theorems established in the
first 4 Sections, would have sufficed to give other analytical ex-
pressions, for the roots of equations, &c. also, in Definite Integrals.

Resolution of Algebraical Equations. 151

SECTION VI.

We shall here add a few theorems, connected with the present
subject.

To transform the definite integral of f (x), taken between two
limits, which are the two roots of given equations; to another, in
which the limits are known quantities.

Suppose (x) = 0, F(x) = 0 are the two equations, the roots of
which, are to be the limits of the /.,f(x); and first, let us consider
the value of this integral, taken from «= 0, to «= the root of the
equation ¢(«)=0; this value is a certain function of that root;

_ ol

and is therefore, by Section (2), = coefficient of a im =

multiplied by the derived function, namely, by f(x); and apply-
ing the theorems given in Section 5, it follows that, /; f(z) is

A 2p pe7oe {o (e*) .c-%
t) QrJf-1

the limits being 0 and 27,/—1.
Similarly, /. f(x) from «=0 to x = root of F(z) =0

seed § P(e), e—°
f) Qarrnf —-1

between same limits.

Hence, if we subtract, we get (f(x) between the two pro-
posed limits, viz.

ff (ey, le) s9=0
(Cea =
02x,/-1* ° p(e') l@=20,/-1

the forms of the functions being such as those already noticed.

152 Mr. Murpeuy on the

In many instances we have had occasion to take the coefficient
of = in f(x), or which is the same, the term independent of x in

a f(x), there may easily be obtained several theorems, for facili-
tating the research of this quantity, the following may perhaps,
from the elegance of its form, be not deemed unworthy of notice,
viz.

The term independent of x in ¢ (« = =)
Bs ; p Sa°h” (x) }” eo (at
= (x)— {xp (a)}’ + aie Suinc moe. &e.,
x being put=1 in the series.

For ¢(«) may be represented by =a,.2”, the symbol =, denoting
the sum with respect to 2,

Hence ¢ (x - *) = 20s. (« - ae

1 baer 1 n Ss
But (« -+) =(1 -=).a + 2)"s
the part of which, independent of «, is

=1—n’+ (* ated) &e.

» 2" -
or = xv" — (« aa ena) &e.

1°, 9°
supposing « to be put=1, after the differentiations are performed.

Multiply by @,, and take the = with respect to x; observing,
generally, that ;

da"

(-G2)_ a (@* $2)
ae £1) Vda ban,”

Sa,d"

and we obtain the proposed theorem.

Resolution of Algebraical Equations. 153

In concluding these few observations on Algebra, we may
observe, that in whatever department of Analysis we seek prin-
ciples, simple in their announcement, and general in their appli-
cations; so far as we succeed, something useful is acquired for
analysis, nor can any branch of Mathematics, however humble,
be deemed unworthy of cultivation in this way, and it is with
this view that the Author has presented this Essay to the
Society.

ROBERT MURPHY.

Carus CoLLEGE,
March, 1831.

ERRATA.
P. 140. 1. 18. put dw" below the line.
— 144. bottom, for f(a) put f(a).
and for f(a) put f(a)-
— 147.45. for V2e—1 put Qn V—1.

Vol. IV. Part I. U

* ; anoiluwmgal lyin Saath tee wsdorsS

Veal, a drtilsalh He pk J
‘ana soot: ih" Oe inc cs

sila tise ti Wi9ttoy ub a :
ot bor fe 2 ait

aang

j *
Sah
Y ra Ge ”
ean '
7 t fy
~ Pay) a”
= oe
. on ;
Cul dae
; ie
be , ie
“ } & 4
i i. 4 : Kae}
eae -
is ae a
ye ha
) ; |
. 2 ¥ a.
a ie
j ~ o ae "
’ ‘YL , : 7 ab! - a "e7
ary fhe
ve P=" i " as ae

V. Mathematical Exposition of some of the Leading
Doctrines in Mr. Ricardo’s « Principles of Political

Economy and Taxation.”

By THE Rev. W. WHEWELL, A.M.

FELLOW AND TUTOR OF TRINITY COLLEGE.
[Read April 18, and May 2, 1831.]

i. Amone a number of those who have recently cultivated
the study of Political Economy, an opinion appears to prevail
that, by the labours of Mr. Ricardo and his followers, a large
mass of our knowledge on this subject has been reduced to the
form of a series of exact logical deductions from a few simple
and evident principles. If this were really the case—if the fun-
damental principles were completely enumerated and clearly esta-
blished, and if their consequences were truly and fully traced—
this branch of knowledge might rightly be considered as having
assumed an exact and scientific character. My present purpose
is not to examine how far these claims are well founded; but to
shew in what manner those portions of the speculations here re-
ferred to, which pretend to such a scientific form, may be ad-
vantageously submitted to mathematical investigation. I have
already, in a former communication, observed that when our ob-
ject is to deduce the results of a few precise and universal prin-
ciples, mathematical processes offer to us both the readiest and
safest method; since by them we can most easily overcome all the

156 Proressor WHEWELL on the

difficulties and perplexities which may occur in consequence of
any complexity in the line of deduction, and are secure from any
risk of vitiating the course of our reasoning by tacit assumptions
or unsteady applications of our original principles. Perhaps _ it
might not be difficult to shew that in several speculations on these
subjects, such errors have not been altogether avoided. And there
is probably a considerable class of readers who will find the doc-
trmes of Political Economy, when put in a mathematical shape,
more clear, compendious, and manageable than in the works to
which I refer. I may add also, that the mathematical formule
which I shall obtain, will be, at the same time, both much more
exact and much more general than the numerical examples in
which writers on the subject are in the habit of embodying and
illustrating their reasonings.

2. I have said that the doctrines of Political Economy have
been supposed to be reduced to a few simple fundamental prin-
ciples; and my business will be to state these principles and
to trace their consequences. The mathematical investigation pro-
ceeds from these principles as postulates, and has no concern
with their truth or falsehood. To extricate such principles from
the mass of facts which observation of the world and of ourselves
teaches us,—to establish the reality, number, and limits of such
laws,—these are offices which belong to a branch of philosophy
altogether different from that with which the mathematician, in
the proper sense of the name, has to deal. Such a task is far
from being either short or easy; and it would be very hasty to
take for granted that it is already completed, or even that any
considerable portion of it is performed. For my own part, I de
not conceive that we are at all justified in asserting the princi-
ples which form the basis of Mr. Ricardo’s system, either to be

Principles of Political Economy and Taxation. 157

steady and universal in their operation, or to be of such para-
mount and predominant influence, that other principles, which
oppose and control them, may be neglected in comparison.
Some of them appear to be absolutely false in general, and others
to be inapplicable in almost all particular cases. Perhaps, how-
ever, to trace their consequences may be one of the most obvious
modes of verifying or correcting them.

I proceed now to enumerate the Postulates which seem to
form the foundations of Mr. Ricardo’s doctrines: and I shall
point out at the same time the manner in which they may be

mathematically expressed and treated.

I. Postulate of Rent.

3. In the system now before us, agriculture is considered as
an employment of capital; and the farmer as a person who lives
on his profits, and who can and will remove his capital to another
employment, if his profits can be so increased, or so prevented
from falling. Hence the farmer will not consent to make less than
the average rate of profits; also, competition will not allow him
to make more: and therefore the excess of the produce of the land
above that amount which is necessary to realize such profits, will
be transferred to the landlord as rent. Hence, on these supposi-
tions, rent is the excess of the produce of capital employed on
land, above the produce of the same capital otherwise employed.

It is also supposed that there are soils of different degrees of
fertility, which form a continuous decreasing series: and that the
lowest degree of fertility on which the cultivator can obtain a
living profit without paying rent, will be cultivated.

It is also supposed that there are modes of employing, on the
same land, successive quantities on doses of capital, and that

158 Proressor WHEWELL on the

each successive dose must universally obtam a _ proportionally
smaller return than the preceding.

Of these hypotheses, the first is generally allowed to be ap-
proximately verified in England: though even here, the reluctance,
difficulty and loss which accompany the transfer of farming capi-
tal to other employments, the moral and social ties which connect
the landlord and tenant, and the numbers of cases in which the
cultivator does not live on profits only, very much limit the
generality, and obstruct and extend the manifestation of the naked
principle thus asserted. And, taking the world at large, it has
been shewn, in the admirable work of Mr. Jones, that this view
and measure of rents is entirely mapplicable, and that none ‘of
the suppositions on which it proceeds have the slightest resem-
blance to the actual state of things. It is to be recollected there-
fore, that whenever the postulate of rent is introduced, the appli-
cation of our reasonings can only be made to cases of farmer's rents :
and that all countries in which serf, metayer, ryot or cotter
rents, or cultivator-proprietors prevail, that is 99 hundredths of
the cultivated globe, are to be excluded from our conclusions.

The existence of a limiting soil, or of a soil cultivated but
paying no rent, is not necessarily implied in the measure of
rent just mentioned. The fact that there are soils approaching
nearly to this limit may however be conceded, and will not, in
most cases, much affect the conclusion. On this supposition,
rent is the excess of the produce of the soil over that of an
equal portion of the limiting soil.

The third supposition, that each successive dose of capital
must procure a proportionally smaller return, appears to be, as

* Essay on the Distribution of Wealth, and the Sources of Taxation. Murray, 1831.

Principles of Political Economy and Taxation. 159

Mr. Jones has observed, an assumption without any @ priori
foundation whatever, and not at all consistent with the known
history of agriculture.

Let r be the number of quarters of corn, grown on an acre
of land, p the price of.a quarter in pounds, e the capital em-
ployed on one acre, including wages; d the sum requisite to re-
place ¢ with the usual profit: then we have the money rent of
one acre =pr—d.

If r, be the produce of the limiting soil, pr,—d=0; and

rent of one acre = pr —pr,.

Since ¢ is the whole capital employed on an acre, it is the
sum of all the doses up to the last.

Instead of a quarter of corn, a pound, an acre, we may take
any other unit of produce, price, and land.

II. Postulate of Wages.

4. Mr. Ricardo assumes that the natural rate of wages is
invariable; that is, that the labourer’s command of food and
other necessaries is never permanently augmented or diminished.
Hence, if the price of corn (or whatever is the main article of
food) rises or falls, a rise or fall in wages shortly follows and
compensates this charge. |

This opinion is supposed to be established by the ascertained
laws of the. progress of population. It is conceived that if the
demand for labour, and consequently the reward of it, is di-
minished, the encouragement to population being thus weakened
a retardation in its advance will occur, which will, after a certain
period, restore the original standard of wages: and that these
effects inverted will occur in the case of an increase of wages.

160 PRoressoR WHEWELL on the

The truth of such a propesition is repeatedly taken for granted
by Mr. Ricardo (3d ed. p. 88. 95. 118. 173, &c.), and is employed
by him as a leading step in several of his reasonings. It would
seem however that the assumption of such a necessary and uni-
versal operation of wages upon population, is entirely gratuitous
and unfounded, and that we have not the slightest ground for
asserting this to be the law of such changes, any more than any
other law arbitrarily assumed.

We might for instance assume that a rise of wages, by im-
proving the chance of the labourer to make some saving, will
increase his prudence and self-control, and thus diminish the
rate at which population increases. Without asserting this to be
more certainly true than the opposite postulate, that a rise of
wages accelerates the rate of increase, we may with reason main-
tain it to be as certainly true m some cases: and this is quite
enough to shew the baselessness of Mr. Ricardo’s postulate as.a
general law.

If the labourers of any country were placed at the bare limit
of possible subsistence, and under a complete privation of pru-
dential restraint;—so that nothing could be taken away from
their means without destroying them by want, nor any thing
added, without bemg speedily absorbed by an increase of their
numbers ;—on such a supposition their condition might be con-
sidered as fixed; and the postulate might be conceded. And
some such supposition, suggested by erroneous views of the laws
of population, appears to have led to the assumption before us.

This supposition however does not, it is to be hoped, repre-
sent the condition of any country. It certainly bears no resem-
blance to that of our own. If we compare the English labourer
with the labourer of France, Ireland, or India, we readily per-

Principles of Political Economy and Taxation. 161

ceive how far he is above the poimt we have described, to which
indeed none even of the others can be said to approximate. To
consider the English labourer as possessed of mere necessaries, in
the severest sense of the word, would therefore be an obvious
and glaring falsity. Hence Mr. Ricardo found it convenient to
take, not the necessary, but the habitual subsistence of the la-
bourer as the permanent standard of wages (p. 91. 95). The
assumption however that the habits of the labourers cannot change
from period to period, so as to accommodate themselves to dif-
ferent amounts of real wages, appears to be quite unsupported by
reason, and in direct contradiction of all known history. The
habitual necessaries and comforts of the labourer may, and do,
undergo changes simultaneous and co-ordinate with those of the
population. Any attempt therefore to derive the fluctuations of
the latter, by supposing the former invariable, is utterly visionary
and unphilosophical.

To obtam such principles as may truly indicate to us the
manner in which alterations of wages do really operate upon the
habits and numbers of the labourers, is an object of great in-
terest and importance, and one in which, though some progress
has been made, much is yet to be done. It seems little likely
that any one general law, free from all control of time, place
and circumstance, will be found of any real use or value.

We may indeed notice, in order to ayoid, the propensity of
the speculative powers of the human mind to rush forwards and
to endeavour to seize on such a general law. The _ professedly
hypothetical statement of a geometrical progression of population,
and an arithmetical progression of subsistence, made by Mr. Mal-
thus for the purpose of introducing his views, has been far more
frequently quoted, than many of the most valuable views of that

Vol. IV. Part I. ».«

162 Proressor WHEWELL on the

eminent philosopher. And if it were to be maintained, for in-
stance, that the rate of increase varies inversely as the existing
population of a given district, or if any other mathematical law
were asserted; though the probabilities against it would be so
strong, as only to be surmountable by an enormous mass of
evidence; the law itself might easily be made to attract notice
with the pretensions of a discovery.

The following is the manner in which the Postulate now
under notice is mtroduced into calculation.

Let w be the wages of a labourer for one year, when the price
of corn is p. Let a fraction f of the labourer’s wages be ex-
pended in corn (or other necessary food), and the remaining frac-
tion 1—f in other commodities which do not rise in price with
corn. Now let the price of corn become p(1+.2). Then the ex-
penditure of the labourer will be (1+2)/fw for corn, and (1—/)w,
as before, for other things. Therefore the whole wages are now .

=(1+a)fo+(—f)w=(1+f2)w.

III. Postulate of Price.

5. The exchangeable value of a commodity depends solely
upon the relative quantity of labour necessary for its production.
Or, in other words, articles which require for their production
equal quantities of labour will exchange for each other.

It is not difficult to shew that all exchangeable value has its
source in human labour; and that this is true of raw produce
no less than of manufactured goods. Mr. Ricardo’s principle
goes further, and asserts that the exchangeable value is in pro-
portion to the quantity of the labour necessarily employed, with-
out any regard to the wages which have been paid; the capital
by which the labour is aided being however estimated as part of

Principles of Political Economy and Taxation. 163

the requisite labour; and the profits upon this capital being al-
lowed in the usual manner.

This measure of exchangeable value is to be understood as
applicable only in the long run, and as the regulating and limit-
ing principle of what may be called the natural price. Under
this restriction its truth depends on the consideration, that if
this were not the law of exchange, some labour would be better
rewarded than some other, and hence the equilibrium would be
restored by a transfer of labourers to the favoured employment.
The principle therefore is true so far as the possibility and ope-
ration of such a transfer extends: that is, it is true of prices in
the same country, and so far as the labour which is embodied
im the commodity is, from conditions of time and place, within
the reach of this competition.

That some kinds of labour are more highly paid than others
does not vitiate this principle; for they are so in virtue of the
education required, the skill of the labourer, or the intensity of
the labour (Rie. p. 11.). And the tendencies which distribute the
numbers and rewards of labourers in different employments, may
be supposed to have reached their equilibrium; so that a change
in the quantity of labour required can alone affect the price.

The principle thus enuntiated is sometimes expressed by say-
ing, that the price must be such as to pay the cost of produc-
tion with profits.

This natural price may often be different from the market
price, which is affected by another principle, that of supply and
demand. According to the principle of supply and demand, the
price increases by an increase of the demand, or by a diminu-
tion of the supply, and vice versa. The exact law which con-
nects these changes of price with those of supply and demand,

xQ
.

164 Proressor WHEWELL on the

has not been ascertained with any precision, and is probably
very different in different cases. When it is necessary to intro-
duce it into the calculation, I shall suppose (as in a former Me-
moir) that the increase of price is proportionally greater than the
diminution of supply.

The principle of supply and demand effects the market price
immediately and at once. If the market price rise, the supply
is increased, and the price thus brought down, and in this man-
ner the market price gravitates to the natural or remunerating
price, which is the standard in the long run.

Let a capital ¢ produce r units of a commodity, p being the
price of one unit. Let 4 be the return which is annually neces-
sary to replace the circulating part of the capital c and the wear
and tear of the fixed part: and let y be the rate of profit. Then
pr=b+ye, is the equation which determines p.

Also if the supply diminish in the ratio 1: 1—y, the price
increases in the ratio 1: 1+ey: (e being greater than 1).

IV. Postulate of Profit.

6. Profits are entirely regulated by Wages.

Wages and Price being determined by the two preceding pos-
tulates, and the power of production of a given capital being given,
the profit left for the capitalist is necessarily determined, being
that portion of the surplus return which is not absorbed by
wages. In this manner profits appear to be an entirely passive
portion of the produce: and when they fall, the capitalist ‘has no
remedy in any possibility of raising prices, or otherwise compen-
sating his loss.

This is true so long as we suppose capital confined to one
country. When, however, profits fall below a certain point, the

Principles of Political Economy and Taxation. 165

capitalist will either cease to retain his capital in its office, and
will convert a portion of it into expenditure: or he will transfer
it to some other country where profits are higher. This resource
is one which, though probably not called into action by a differ-
ence of one or two per cent. between domestic and foreign pro-
fits, will undoubtedly become very active and influential, if the
difference should go much further, and will operate by dimi-
nishing the supply of the capital at home, till profits become of
such a magnitude as to be a sufficient inducement to the em-
ployment of capital at home.

I shall suppose + to represent the profit on £1; so that if

protits be 10 per cent., y is a or=, 1; if 15 per cent., y=,15;

if 30 per cent., y=,3. The rate of profit will be 100 per cent.

V. Postulate of Equilibrium.

7. In the reasonings which we have to follow, the principles
which we have stated are supposed to be carried into complete ope-
ration, and commodities to be distributed according to the laws
to which those principles give rise. Thus the rate of profits of
agriculture and of other employments are assumed to be equal,
it beg supposed that capital is tranferred from one employment
to another, till such an equilibrium is produced: though this is
a process which manifestly would require a considerable time,
and would never be completely performed. In the same man-
ner, assuming the postulate of wages (which as I have said is
perfectly gratuitous) it is clear that the changes of population
which it supposes would require for their accomplishment a
considerable course of years, during which new causes and cir-
cumstances might come into action, so as entirely to modify the

166 Proresson WHEWELL on the

result, even if the tendency of the original cause had been rightly
stated. In the postulate of price, it has already been noticed,
that the market price, determined by the immediate action of
demand and supply, may be very different from the natural
price, determined, according to the postulate, by the cost of pro-
duction : this latter price being however that under which the equi-
librium obtains, and to which the other perpetually bends.

Supposing the preceding postulates true, the problems in which
they are applied are much simplified by assuming such an equili-
brium to obtain: but along with this simplification we incur a ne-
cessary and perpetual, and, it may be, a very considerable deviation
from the circumstances of actual fact. In reality, this equilibrium
is never attained: probably in most cases it is never approxi-
mated to. There is a constant tendency towards the state of
things in which the elements of wealth are in this exact balance,
but this is a tendency like that which the waters at the source
of a river have to descend towards its mouth. We cannot from
such a tendency infer that the whole course of a river is at
the same level; and just as litthe may we flatter ourselves that
we have solved the problem of the course and distribution of
the current of wealth, when we have combined the laws according
to which an exact balance might be produced.

We are to recollect therefore, that even if our principles
were exact, deductions from them made according to the method
we are now following, would give us only a faint and distant
resemblance of the state of things produced by the perpetual
struggle and conflict of such principles with variable cireum-
stances. Such deductions however would probably have some
resemblance, in the general outline of their results, to the true
state of things. They would offer to us a first approximation :

Principles of Political Economy and Taxation. 167

and in difficult problems of physics, it is precisely by such a
simplification as this, that a first approximation is obtained.
Thus in the investigation of the problem of the tides, we have
a very complex case of the motion of a fluid: but Newton’s
mode of treating the question was, to consider what would be
the form of equilibrium of the ocean, acted upon by the forces
which produce the tides: and this solution of the problem, though
necessarily inexact, was accepted as the best which could easily
be obtained. The investigations of Laplace and others who have
since treated the problem on its true grounds, as a question of
hydrodynamics, have shewn that Newton’s solution explains rightly
the main features of the phenomenon.

In order however that solutions of this nature may have any
value, it is requisite that the principles, of which we estimate
the operation, should include all the predominant causes which
really influence the result. We necessarily reject some of the
circumstances and tendencies which really exist: but we can do
this with propriety, only when the effects of these latter agents
are, from their small amount or short duration, inconsiderable
modifications only of the general results. The quantities which
we neglect must be of an inferior order to those which we take
into account; otherwise we obtain no approximation at all. We
may with some utility make the theory of the tides a question
of equilibrium, but our labour would be utterly misspent if
we should attempt to consider on such principles the theory of
waves.

It appears to be by no means clear that the irregular fluctu-
ations and transitory currents by which the elements of wealth
seek their natural level may be neglected in the investigation of
the primary laws of their distribution. It is not difficult to con-

168 Proressor WHEWELL on the

ceive that the inequalities and transfers produced by the tem-
porary and incomplete action of the equalizing causes, may be
of equal magnitude and consequence with those ultimate and
complete changes by which the general tendency of such causes
is manifested. A panic may produce results as wide and as im-
portant as a general fall of profits.

Of this source of uncertainty in the value of such results as I
shall have to obtain, I cannot pretend to determine the extent.
But independently of this consideration, the postulates above
stated are so far from being adequate properly to represent the
general facts, that we can hardly look for any accordance
between the calculated and the observed effects. To consider,
for instance, the tendency of mankind to increase their numbers
as a universal and inevitable law, and to leave out of consider-
ation the co-ordinate and antagonist tendency by which they
endeavour to preserve and increase their comforts, is to insure a
total dissimilarity between our theory and the actual state of
things.

The postulate of equilibrium is introduced to our calculations
by the process of putting the other postulates into the form of
equations. The values of the quantities involved are by this
means determined according to the condition of the equilibrium
of our principles above stated.

8. I will now mention the general problems which it appears
to have been the object of Mr. Ricardo’s work to solve; and I
will afterwards proceed to obtain mathematical solutions of them
on his principles.

If we suppose the population of a country to go on increasing,
and the powers of agriculture to remain stationary, it will, in order
that the increased numbers may be provided with subsistence, be

Principles of Political Economy and Taxation. 169

necessary that more and more capital and labour should be ap-
plied to the task of raising food. If we suppose these to be em-
ployed on new land, more and more land will be perpetually
cultivated: and if we make the supposition already mentioned,
that the land of the country consists of a progression of soils of
decreasing fertility, each new soil cultivated, will yield a less sur-
plus produce to the labour employed upon it.

This increase of population, and consequent extension of agri-
cultural labour to less productive soils, Mr. Ricardo conceived to
have been the progress of things in this country: and apparently
he conceived it also to be the necessary and universal progress of
nations. On this supposition his first main problem was to trace
the distribution of the various portions of the produce, as wages,
rent, and profits, which takes place in the course of this progress.

That this has been the course of events in England seems to
be clearly and demonstrably false. After Mr. Jones’ reasonings
(Essay, chap. vii. sect. 6.), I do not conceive that any doubt
can remain on the subject. And it is remarkable that Mr. Ri-
cardo’s error in this instance is not a mistaken assumption of
principles, but it is a defect in his deduction from his principles,
a part of his task which is generally supposed to be unexcep-
tionable. The error resides in his having neglected altogether the
effects of an increase in the power of argriculture, which, in Eng-
land, has been a change at least as important and as marked, as
the increase in the population. This being the case, it is evident
that the whole of his assumption of the nature of. the econo-
mical progress of this country, and the views of the distribution
of wealth arising from this assumption, must fall to the ground.
It may however still be curious to see the exact consequences of
the assumptions now referred to; and moreover, our formule

Vol. IV. Part I. ve

170 Proressor WHEWELL on the

will be applicable with no great modification, to the case in which
the powers of production are supposed to have increased.

Mr. Ricardo’s other problems are these.

In the course of the progress of things above described, taxes
being levied on any portion of the wealth of the country (as
wages, rents, land, profits, &c.) to determine their ultimate inci-
dence and consequences.

Mr. Ricardo has also considered some of the questions con-
nected with the subject of foreign trade and the varying value
of money, which I shall afterwards endeavour to state.

PROGRESSIVE DISTRIBUTION OF PRODUCE.

9. Let the following notation be employed ;

7 =the produce on one acre, p =the price of one quarter, ¢ the
capital on one acre, y the rate of profit on £1, 7 the number
of labourers employed on one acre, (including the labour requisite
to replace the waste of fixed capital) w the wages of labour.

The value of the return on one acre is pr, of which the
labourers’ portion is 7w, and the capitalist’s is yc; hence by
Post. I. pr—lw—vye is the rent. On the limiting soil this gives
pr=lw+ye. Now suppose prices to rise to p’, and cultivation to
be pushed upon a worse limiting soil, on which the produce is 7’,
the capital ¢’, and the labourers 7. Hence p'r’=/w'+y'c. And
in this case the rent on one acre of the former limiting soil be-
comes p'r —lw'—y’c.

The increase of price is supposed to be in proportion ‘to the
increase of labour requisite to produce the same quantity of corn
(see Post. III). The whole price pr was, in the first instance,
paid for the result of the labour 7; the whole price p’r’ is after-
wards paid for the result of the labour 7; therefore pr: pr’ 7: 7..

Principles of Political Economy and Taxation. 171

On the second limiting soil

one Vel rt ’ f ‘Tw
pr =lw'+y'c; whence vite
ie (ji) , , —l ’ ii
or since prl=prl’, + ee

Hence y'c=(pr—Jw’) = :

Let the price increase, so that p'=p(1+); then (Post. 11.)
w =w (1+f2).
Also let pr=nxlw. Then we have, for one acre of the first soil,
after the second is cultivated,
Wages =lw' =(1+/2) lw.

Profits=y'e= jnlw—(1+f2) lw)} =
=(n—1—f2r) tw.

Rent=p'r—/w' -yc
=(1+2) nlw—(1+f2) lw—y‘e
= fn —1 tne fe—(n=1=fe) Sh
These are the amounts in money. The amounts of .these
quantities in corn will be found by dividing each by the price
p =p(1+2).
If we suppose the capital on an acre of the new soil to be

greater in proportion as the labour employed upon it is less, we
have c/’=cl: and in this case, for one acre.

Wages=(1+/2) lw.
Profits =(z—1—fx) lw.
Rent =nalw.

¥2

172 ProressoR WHEWELL on the

The amount of capital and labour which will be employed on
an acre of the new soil will depend upon its agricultural consti-
tution. The farmer will fix upon such quantities as give him the
most profitable returns.

In the case supposed by Mr. Ricardo, p. 115, 7=/, c=c. Hence
the last formule apply; and we haye for the portions of the
value of the produce

Ret. Profits. Wages.
at first, price = p, 0 (n—1)lw lw
afterwards, price = p’, nalw (n-1— fx)lw (1+ fr) lw

: 2 x n—1— fx 1+f2
fractions of the whole = 57 ma ima ise ay maces

the whole produce is, in the first case, in money pr or xlw, and
in corn 7: in the second case it is in money p’7 or (1+2)nlw,
and in corn 7, as_ before.

We had /= (P7= WF. similarly y =2°—";

(b
y Be area ey Cr I, BS
therefore re a 7:

In the case taken by Mr. Ricardo, as before, cl’=cl. Also

1
ui; n=3; fe = Io?

1
therefore y'= a x ,16=,152.

From these formule we may obtain the numerical results
which Mr. Ricardo has given at p. 115. (See also pp. 75, 98, &c.).
He supposes 7 te be 180 and 7’ to be successively 170, 160, 150,
140; f is 5: Also, in the examples given, »=3. If we make,

for instance, 7’= 150, we have

orale best w= w(1+ =)
150" VBIT. G2 Ti 10

Principles of Political Economy and Taxation. 173

9
36’ 36°
being 180, these are 30, 95, 55 as in Mr. Ricardo’s example. Also

and the three fractions of the whole are m The whole

zeprinece,. (1)
the price is 2P where p=£4: whence the money value is easily

calculated.
Since the sums paid as wages and the rate of wages differ
at the two periods, the capital employed will differ. If, how-

ever, we suppose the value of the capital the same at the two

n—1— fx
n—1

If the capital be supposed to be agricultural produce and con-

periods, the rate of profit is altered in the proportion

sequently its value to partake of the increase of prices, profits
n-1— fx

(n—1)(1 +2) Compare

would be diminished in the proportion
Ricardo, p. 116.
PRICES.

11. In the above example, Art.9, the price of corn is sup-
posed to depend entirely on the labour directly employed to
produce it. This is Mr. Ricardo’s mode of treating the sub-
ject. It is, however, somewhat inaccurate according to his own
principles: for in applying the postulate of price, we are to take
into account the capital employed as well as the labour.

In consequence of this consideration, the prices of commodities
will be effected by the proportion and durability of fixed capital
requisite for their production. Mr. Ricardo has several prepo-
sitions on this subject which may be included in the following
formule.

Let 7 be the number of labourers who this year work by
means of a machine or any other kind of fixed capital: / the
number of labourers who were employed last year in making

174 ProressoR WHEWELL on the

this machine: y the rate of profit. The machine at the end of
last year was worth the wages expended on it, together with
profits, that is (1+-)/w. It must produce, this year, the profit
on what it is worth, and the wages of those who work it, with
profits: that is, it must produce y(1+-y)lw+(1+ +7) lw.

In the same manner let a machine have been produced by
the labour of 7 labourers last year, 2’ in the preceding year, 7”
in the year before that, and so on. And let / men be employed
in working the machine; then its produce must be worth

(1+-y)lw+(L++y)Tw+ (1+ y)?l'w + &e.

Let L be the number of men requisite to obtain the same
produce without the machine. Then Lw is their wages, and (1+7)
Lw the value of the produce. Therefore, if the machine can be
employed without loss

(l+y)lw+(l+y)*lw + (i+y)°l'w+&e.= or <(1+y) Lu,
and 2+(1++7)%+(1+-y)'2’ + &&.= or <L.
Hence /4+7+4+/'+&.<L;

or, when machinery is employed, it has always cost less labour
than would obtain the same produce without machinery, Ric.
p. 44.

Let 6 be the sum expended upon an average, every year for
k years, in the construction of the machine, @ the value of the
machine when constructed. Then
sty 9S i i

12. When a machine, or stock of any kind, is employed in

e=bi(l+y)+(1ty)*-- (+ yy

production, a certain sum is annually requisite to replace the
wear and tear of the capital so existing. Mr. Ricardo supposes
these annual sums to accumulate at interest, so as to become suf-

Principles of Political Economy and Taxation. 175

ficient to purchase a new machine just when it is wanted. Let
& be the duration of the machine; 6 the annual reserve for it;
therefore we must have an annuity } for & years, of which the
present value is c. Therefore

Be oe ey
(l+y)'-1 1-(1+¥y)7*"
C 1 ¥Y 1 1
SN Sa
se ae aan TO yy yy a

3 1 ' 3
As & increases ya+yy decreases, and d increases. For instance,
z

a 10
it 1 => when £=1, d= —7> and when 4=10, d=6,14.

Now we have
pr=lw (1+7)+b+ye,

=lw (1+-¥) +5+ yc:

or if c=mlw,
m
pr=lw j\1l+y+t+ at ym):
Let wages rise to (1 +«)w: and let p become in consequence
(1+2)p; then

(1+2) pr=lw §(1+¥7’) (1+w) +5 +m}

subtracting, 2pr=lw {u(1+-) —(y—y') (1+ m)} ;

And it is manifest that wv being given, and y thence deter-
mined, the value of « will be least in the cases where m is the
greatest; that is, the rise of price is greatest in the cases where
the proportion of fixed to moveable capital is greatest.

The rise of price is here estimated in labour. But if we sup-
pose money to be produced always by an equal quantity of

176 Proressor WHEWELL on the

unassisted labour, we may measure by it the variations of price
in other things.

Let » labourers, unassisted by capital, produce p units of
money. When wages rise, the produce of the same labour will
be of the same value, therefore the unit of money will not alter
in value. Hence we have

p=Aw(1+y), p=Aw(1+u)(1+y)

, , UES,
=I+>5: — .
(l+y)(1+u)=1+y; l+y iu:
1
yy hence by what precedes

aph=etr — (1 Ha ” pete ty)
This is always negative, and hence we have Mr. Ricardo’s as-
sertion that on these suppositions, prices fall by a rise of wages.
(p. 41).

The above calculation is inexact, for d would alter when
becomes y': but if d be considerable, this error would be in-
considerable.

The supposition here made, that profits are entirely passive, is
manifestly inadmissible; for if profits were 10 per cent., a rise of
10 per cent. in wages would annihilate profits altogether: which
is absurd.

TAXES.

13. The general problem on this subject, is to determine the
ultimate incidence of any given tax, according to the preceding
postulates.

(1). Taxes on raw Produce (Ric. Chap. viit).

Let a tax which is a portion ¢ of the whole, be imposed on

the produce. It affects all soils, and therefore the limiting soil,

Principles of Political Economy and Taxation. 177

on which the produce is 7. Let the price before the tax be p
and after it p(1+2); the rest of the notation as before. Hence
(1—#)(1+2)pr is now the value of the cultivator’s portion of the
produce, which must pay wages and profits. Also profits remain
unchanged, for if they were to fall, the cultivator would remove
his capital to other employments. Hence

pr=lw+ye, pr(l—d(1 +2) =lw(1+fx)+ye
subtract, and pr(1—d)(1 + «)—pr=lw.fx: whence

aes prt ov nt
~ pr(l—d — flw ~ n(1—t)—f*

1 : 1 3 3 F
If, as before, n=3, Jigs and if iy: =55; prices rise as 11:14.

In these cases prices rise so as to compensate the whole of the
cultivator’s loss and of the increased expenses of the labourer.
The whole tax (assuming our Postulates) falls upon the rich
consumer,

This is true on the supposition that the consumption is not
at all affected by the price. Col. Thompson has properly ob-
served that this assumption is by no means correct; and has
shewn that if we make the more natural supposition that an in-
crease of price will diminish the consumption, Mr. Ricardo’s con-
clusions are no longer true. Pursuing Col. Thompson’s views, I
have, in a preceding memoir, given formule expressing the por-
tions of such a tax which fall respectively in Rent, Profits, and
Prices, and have shewn that in general the former is the largest
portion. (Camb. Trans. Vol. 11).

According to the postulate of wages, no part of such a tax

could fall on wages.
Vol. IV. Part 1. Z

178 Proressor WHEWELL on the

The postulate of wages is however, as I have already said,
perfectly gratuitous; and there does ‘not appear to be any good
reason for believing that a tax upon raw produce, whether food,
or any other articles, might not fall to a great extent on the
wages of the labourer.

The effect of a change of wages, such as is here supposed,
upon foreign trade, will appear from what we shall have to say
in treating of that part of the subject.

2. Taxes on Rent, (Ricardo, Chap. vit.)

If be the produce of the limiting soil, and 7, of any other
soil, we shall have, after the tax as before, pr=/w+vy: and as y,
r, c, J, w, remain the same, p is not altered. Hence if a tax is laid
on rent to the amount of the fraction ¢, it will fall wholly on
rent: which will be reduced from p7,—pr to (1-4) (p7,—p7).

If what is called rent, be in part interest for capital, a tax on
rent will be governed by different laws, so far as that portion
is concerned.

3. Land Tax, (Ricardo, Chap. x.)

Let 7’ be the amount of the tax per acre. Therefore pr’ —T
is the cultivator’s portion of the produce of the limiting soil;
y will not be changed; therefore

pr-T=lw'+ye, pr=lw+ye,
—_— T _ T .
~ pr—flw (n— ftw’
the tax will wholly affect the price, and the results will. be ex-
actly the same as in a tax on raw produce.

(p—p)r—-T=lw'—w); pre-T=lufx; x

4. Taxes on Profits, (Ricardo, Chap. x11.)
These taxes are supposed to be partial, that is, to be levied
only on certain employments of capital. In that case they will

Principles of Political Economy and Taxation. 179

not diminish the rate of profits, because if they did, the capital
would be transferred to other employments.

As before, let » be the number of units produced by a total
capital c, (fixed and circulating), p the price of each unit. Then
pr=lw+ye: profits are ye. If the tax be a fraction ¢, y will
remain the same after its imposition. Let the price become jp’ ;

therefore
pr=lw+ycitye.
y t
Hence 2 =1 + —”% i
p lw+ye

the productions of those employments of capital in which profits
were taxed, would rise in price, compared with those in which
the tax was not applied. If money be not taxed, the money
prices of such commodities will rise.

It has been seen, (Art. 12.) that if a person employ a fixed
capital c—lw, which is replaced in & years, and a circulating
capital /w in paying wages, we must have

pr=lw+b+yc, where ba
in this case profits are yc.

If we now suppose profits to be taxed, and money not to be
taxed, money prices will rise. Mr. Ricardo supposes that in this
case the money value of the profits will remain the same as be-
fore, (p. 272. Ist ed.).

Therefore p'r=lw+b+ye+tye;

pry, re meas Ls
"p lw+b+rye
Let the fixed capital be m times the circulating, or e—lw=mlw;

mlw

d
greater as / is greater (see Art. 12.), then

hence c=(m+1) lw. Also let b= , where it is clear that d is

180 Proressor WHEWELL on the

Pw ti py soe!
P 1+ 9 + (m+1)y

atte f —.

ry (m +1)

Hence it appears that the increase of price on these suppositions

will be greater according as 1 +4 is less.

1+ m™
1+5 1-5
But —— =1 — ——_.-
1+m™ 1
1+=
m

Therefore (d being >1) the rise of price from the tax will be
ereater in commodities, as m is greater, and as d is less; that is,
as the proportion of fixed capital is greater, and as its dura-
bility is less, (Ricardo, p. 273).

If we suppose two cases, in one of which »=4, in the other

5 5 v] G
m =5 (Ricardo, p. 273.), and if we neglect a we have, y being

A
5?

in the first case 2 = 1 +5

in the second - =1+

The rise of price is 24 times as great in one case as in the
other.

A tax on all profits is an income tax, and the effect of such
a tax will thus be unequal on the prices of different commo-
dities.

Principles of Political Economy and Taxation. 18]

The preceding calculation is upon the supposition that the
money metal is produced in the country in which the tax is laid.
If the precious metals be imported from other countries, the in-
vestigation of the consequences will require considerations which
will hereafter be stated in treating of foreign trade.

5. Taxes on Wages. (Ric. Chap. xtv.)

We assume still the postulate of wages, although, as has been
said, it appears to be unfounded, and consequently the infer-
ences to which it leads, as to the incidence of taxes, will not be
verified in fact. But, granting the principle, there appears to be
some difficulty in tracing its results, and Mr. Ricardo, in attem pt-
img to do so, has been led into an arithmetical fallacy, as was
pointed out by Col. Thompson, and as I have shewn in a pre-
ceding Memoir. (Ricardo, p. 301, and Camb. Trans. Vol. 111.
p. 192). Mr. Ricardo’s opinion was, that taxes upon wages fall
entirely on profits, while other writers maintain that they will affect
prices and rent. Calculations such as we have already employed
will enable us to obtain the true result in this case.

Let the labourer expend a fraction £ of his wages on agricul-
tural produce (as corn), and the remainder 1—f on manufactured
goods. And by the postulate of wages, let his real consumption
remain the same after the imposition of the tax, as it was before.
If prices rise, the rise may be different in corn and in goods,
according to the proportion of fixed and circulating capital in
the two cases.

Suppose that for corn the fixed capital c—/w=mlw: for goods in
the same manner let »rAw be the fixed capital: and let the wear
and tear of the fixed capital be included in the circulating capital.
Also suppose that after the imposition of the tax, the price of

182 Proressor WHEWELL on the

corn rises in the proportion 1:1+2, the price of goods in the
proportion 1:1+£€; and that wages rise in the proportion 1:1+w.
Therefore the wages of the labourers employed in producing corn
become /w(1+z); and we have by the condition of remunerating
price,
pr=lwl+y)tymlw;
(1+ 2) pr=lw(1+u)(1++7’')+y'mlw; whence

(1+w)(1++’) +-y'm

1 +? Tea similarly
(L+w)(1++) +y'u
1 fe er
f lt+yt+yu

Now by supposition the labourer’s consumption of corn and
goods is the same as before; and therefore the expense of corn,
which was fw, becomes f(1+2)w. Similarly the expense in goods,
which was (1-)w, becomes (1—/)(1+£)w: and these portions
together with the tax make up the whole of the wages (1+2)w.
Let the tax be a portion ¢ of the whole. Then

J(1+a)+(L—f)(1+é)+¢#(1+u)=1+4, or

fA+WA+y)+fy'm , I=f\a+w)(L+y)+0 —f)y'n
l+yt+ym l+yt+yph

= (1—A)(1+w).

From this equation y' being known, uw is known, or vice versa.

It appears from the result that the problem of the incidence
of a tax on wages, under the conditions here supposed, is inde-
terminate. A part falls on the consumer in raised prices, a part
on the capitalist in diminished profits; and the principles hitherto
assumed do not enable us to determine the respective amount of
these portions. In fact we suppose both the consumer and
the capitalist, to be entirely passive, so as to have no power of
throwing their loss on any other person, and on this supposition

Principles of Political Economy and Taxation. 183

it is manifest, that there is nothing to disturb the equilibrium,
whatever be the amount of this loss in either case.

It will be seen however, that the result which Mr. Ricardo
appears to have inferred, viz. that the whole tax in each case
would be taken from: profits, cannot possibly be true. For im
such a case it would differently affect profits in different employ-
ments, according to the proportion of fixed capital: and hence
some succeeding adjustment must necessarily take place, in order
to bring back profits to an equality.

1. Let it be supposed that profits are unaltered by the tax.

Therefore
(l+y) zu

Ya a SL eatyme

wis (l++y)u gel (_+y)4_
l+ytym’ l+ytye

Also f(1+2)+(1—f) (1+%)+¢(1+u)=1+4,
gives ¢=(1—t)u—fx-(1—f)é

= {1-r- f(+y) Ga A+y} y
1+y(m+1) l+y (tl) J ©

whence uw is known.

If, as an extreme case, we suppose no fixed capital in agri-
culture, and no circulating capital in manufactures, m=0, n.=0,
whence

t
t=u, €=0. futi+tu=u, vu =———..
E fut U, u ey,
1 1 1 1
(A = at i= = c
If 10° Ff. gq we have u =F: A tax of 19 On wages would

raise the price of agricultural produce +, while manufactures re-
mained unaltered; and would raise wages in the same proportion.

184 Proressor WHEWELL on the

2. Let it be supposed that prices in any particular employ-
ment, as manufactures, are not effected by the tax; therefore
==0. Therefore

(l+w) (1+y')+yu=l+y+yu; hence
Gy) (e+ 1)
Pinos en
(+y)u(u—m)
(ut+1+u) (1+y+ym)
And the equation f(1+a)+(1—/) (1+&)+¢(1+u)=1+4, gives, m

this case, fr+t+tu=u, whence

Pa (ca a9 a ,
epee rrnEe k ans

Hence, substituting, « =

t=u—tu—f2

S(L+y) (u=m) hy

oF t= fie - (l+y+ym)

whence ~ may be determined by a quadratic, and hence y.

1 1 1
Let ae p=Q, Y=Jo: ier

t=41—7¢ worl"
-| ~~ 46 (8 +u))

3. Let the proportion of fixed to circulating capital be, on
the average, m, so that n=m, E=2.
Then f(i+a)+(1—f)(1+a) +40 +4) =1 +4
or ¢+tu+r=u

u(1+y’)—(m+1)(y-7)
ltytym

=U.

t+tut+

If we suppose, moreover, that prices remain the same after
the tax,

t
2=0, t+ifu=u, “= i

Principles of Political Economy and Taxation. 185

/ 1)y—u
Iso u(i1+ —(m+il inp ie (m+ V)y—u
A (l+7)—(m+1)(y-y')=0; ¥ Ses Ea

i apeeay , dae 1
If trap u=a3 a tax of To Taises wages 9°

Also for the effect on the rate of profit, we have

profits are less affected, as (m) the proportion of fixed capital, is
greater. ,

FOREIGN TRADE.

14. We now come to a part of the subject altogether distinct
from that which we have been considering; a part both of great
interest and great difficulty: I refer to the doctrines concerning
foreign trade; and, as connected with these, the laws of the in-
flux and efflux of the precious metals, and the influence upon
prices exercised by their abundance or scarcity. This is a por-
tion of Political Economy on which the postulates which have
hitherto been the basis of our reasoning have no bearing. The
proportionality of the exchangeable yalue to the cost or labour
of production no longer obtains, when the labour of. different
countries is concerned. “The produce of the labour of 100
Englishmen may” as Mr. Ricardo says, “be given for the labour
of 80 Portuguese, 60 Russians, or 120 East Indians.” Nor can
we assume the equality of profits in different countries. The
difficulty with which labour and capital travel from one country
to another, is a sufficient obstacle in. the way of the establish-
ment of such a uniformity of the value of labour, and of the rate
of profits.

Vol. IV. Part I. Aa

186 Proressor WHEWELL on the

I shall have to state some of the principles which are said to
apply to the questions now before us; they have been put for-
wards with great ingenuity by Mr. Ricardo and his followers; and
several curious speculations have been founded upon them, espe-
cially those connected with the influence of the manufacturing
skill of a country upon the influx of gold into it, and conse-
quently upon the general scale of prices.

It will be seen, that besides the principles which I borrow
from Mr. Ricardo, I shall have to make several assumptions for
the sake of reducing the problems before us to calculation. It
will of course be understood that these assumptions are not in-
tended to be maintained as exactly, or even as approximately
true. Their use is to shew how the numerical examples which
serve as illustrations of the principles, may be properly and con-
sistently generalized and limited. But though these investigations
cannot pretend at present to much precision, it is hoped that
they may serve to shew, of what kind and how many are the
data on which the exact solution of such problems must depend :
and they may thus be of some use in directing future attention
with regard both to the laws and the quantities involved in these
difficult questions.

I suppose the Commerce of different countries to take place
free from all prohibitions, duties, drawbacks, &c. This simplifi-
cation appears to be the easiest mode of undertaking .the inves-
tigation at first. I may, perhaps, on some other occasion con-
sider similar problems with the additional conditions which such
restrictions introduce.

On this supposition we shall assume the following principles.

Principles of Political Economy and Taxation. 187

POSTULATES CONCERNING TRADE.

IT. The merchant will buy a commodity at one place, and
sell it at another, if by so doing he can recover the money cost
with the profit usual in his own country. The consumer will
buy a commodity of foreign rather than the same commodity of
domestic production, if by so doing he can obtain it cheaper.

This is the fundamental principle of all foreign trade, all
restrictions being, as has been said, supposed to be removed.

II. When a country has as much gold (or any other money-
metal) as is wanted to circulate her commodities; if an additional
quantity of gold be imported and retained in the country, the
circulation of commodities remaining the same, money prices will
rise.

Gold being more abundant, will of course become less valu-
able; that is, commodities will be more valuable, if compared
with gold, or gold-prices will rise.

[ shall suppose, in what follows, that prices rise in proportion
to the increased quantity of gold: that is, if g be the original
quantity, and 4 the quantity imported, prices will rise in the ratio

&§:gth. If p be the price at first, p (1 + 2) is the price at last.
5

If however gold is imported in consequence of a balance of
exports over imports, it is probable that the internal trade would
increase, and the circulation of commodities in the country would
be greater than when there were no exports or imports. Hence
a greater quantity of gold than g, the original quantity, would
now be wanted, and the price would not rise so much as has

F P h , ?
just been mentioned. Instead of p(t - =) it might become
5

AAQ

188 Proressor WHEWELL on the

p(t 7 a) when g’ is the quantity of gold requisite for the in-

ternal circulation of the country in its new condition. If we
suppose g’=g+mh where m is a fraction; the price at last

, h g
cp'=p (1 ha) =) + 2)

1l+m—
s

h h.
Or more generally we may suppose p'=p (a +2); when ~- is
5 fo)

some quantity depending on *

It appears to be very difficult to find any criterion or measure
of the quantity of gold requisite for the independent interior cir-
culation of a country: consequently it is extremely difficult to
ascertain the effect on prices produced by the country having
more than this quantity, which, as will be seen, is the result of
a superiority in manufacturing skill.

It is manifest, however, that the quantity of gold requisite for
circulation in any country is diminished by any contrivance which
introduces any other circulating medium, as paper; or which in-
creases the velocity and facility of any portion of the circulation.
The value of the paper (if convertible) depends immediately on
the value of the gold; but the quantity of gold wanted, may
depend on the use and quantity of paper; and the quantity of
gold wanted in a country is one of the conditions which affects
its value there relatively to that in other countries.

If a part of the currency of the country be paper, the effect
on prices produced by the introduction of superfluous gold will
be more difficult to estimate. For the sake of simplicity how-

Principles of Political Economy and Taxation. 189

ever I shall still suppose the effect proportional to the quantity
introduced. If g, be the money at one period, and g, the same
with the addition of the imported gold, I shall suppose that
prices increase in the ratio g, : g.

The preceding considerations may seem to shew how difficult
and complex must be all calculations on this subject.

Ill. In the condition of equilibrium of a trading country,

the annual exports and imports must be equal in money value.

We speak here of a condition of things which may continue
from year to year unaltered. It is clear that in such a condition
we could not have a regular excess of exported value over im-
ported, or the reverse. For if we had the former case, the
balance must be brought here in gold, and we should import
a fixed quantity of gold every year. But in consequence of
this circumstance, the prices of all our commodities would rise,
some that had been usually exported would cease to be so from
their increased price, and this operation would never stop till
the exports had reached a point at which their money value
was equal to that of the imports.

Let p be the price of a unit of any exported article, e the
number of units exported: similarly let p’, p’, &c. be other prices,
é, ¢’, &e. the corresponding quantities: then ep+e'p'+ e’p’+ &e. is
the value of the whole export. This may be represented thus
Sep. In the same manner if ¢, 7, 7’, &. be the quantities im-
ported of articles of which the prices are s, s‘, s' Xe. is+is'+0"s"
+&e.=Sis represents the whole of the imports. Hence we have
by the principle just explained Sep = Sis.

IV. The prices supposed here are the remunerating prices
in the equilibrium of demand and supply.

190 Proressor WHEWELL on the

Let p be the price at which cloth can be imported into Chili
from England, with English profits. Then it is here supposed
that p is the price at which it is there sold, though it may for
a time be sold for a higher price from the demand being active,
or for a lower price, the supply being excessive. The gradual
extension of demand, and the fluctuations and miscalculations of
supply, though causes of perpetual and powerful operation, are
left out of consideration, as not effecting that condition of equi-
librium to which our investigations here refer. The estimation of
these alternations, indeed, appears to be a fitter employment for the
sagacity of the practical merchant, than for the reasoning of the
theoretical economist. To apply mathematical formulz to them
would probably not be a more successful undertaking than it
would be to calculate on mechanical principles, the alternation
of smaller and larger surges in the progress of a rising tide.

15. I shall now proceed to determine mathematically the con-
ditions and consequences of an export and import trade.

Let p be the price of a unit of a commodity manufactured
here and carried into a foreign. country, p' the price at which
it could be manufactured in the foreign country: then if p<p'
the article will be exported. Let e be the quantity exported,
then ep is the value exported of such a commodity: and Sep,
the sum of all such products, is the whole exported value.

Similarly, if s be the value of a commodity imported, s' the
price at which it could be manufactured here, it will be im-
ported if s<s. And i being the quantity, Sis is the whole im-
ported value.

If Sep is not =Sis, the balance Sep—Sis must be imported
in gold, which will thus be added to the quantity of gold

Principles of Political Economy and Taxation. 19]

existing in the country. Hence prices will be altered, and with
them the quantity of exports.
Let ¢,, ¢,...e, be the exports in the Ist, 2d...¢'" year,

Pw» Pe---P» the corresponding prices,

21 Sg. the quantity of gold (or other currency) in the
country at the beginning of these years, (the amount of the cur-
rency being supposed to change only in consequence of the im-
portation of gold).

Also let it be supposed that the increase of price is in pro-
portion to the increase of the quantity of gold, (see Post. IT.;)

that is, » =e
51

We have therefore, since g:,,—g, is the gold imported in the ¢”
year, which must equal the excess of exports, Se.p,—Sis=g,,,—2,
or, putting for p, its value,

PiSt .
—— Se,—Sis=21,.—815
1

therefore gi,, = (1+ P: Se.) g.—Sis...(1).
Si

If we know the law of the diminution of the annual exports
Se, we have Se, in terms of ¢, and hence, by integrating this
equation of finite differences, we can easily find g, in terms of
t, and hence the quantity of gold in the country after any num-
ber of years. In practice (on the suppositions already made) ¢;
would tend rapidly to a limit, and the quantity of gold after a
few years would receive no sensible increase. If ¢ be the num-
ber of years which gives the limit with sufficient accuracy, we
have

Sep,—Sis=0;

" Sis...(2).

, eiSe.— Sis=0, Pp Sse=

ie,

192 ProressoR WHEWELL on the

16. It will be more simple to suppose the successive quanti-
ties of exports Se, Se, Se, &c. to be known. In this case the
calculation will stand as follows.

Let 6,21 ee he
&. &

1
namely, the ratio of the gold originally in the country + the
value of the exports in the ¢" year, to the original gold; the
exports being reckoned at the original price. Also, let the ex-
ports =2g,. Then we have, by equation (1)
St41 = MS: — NS,
which may be easily reduced to calculation when the number of
terms m,, m,, m,, &c. is not large.
Let the exports and imports reach their equilibrium in three
years; we then have the equations
82=M,8i—N8,
23=M.g,-—Ng,
84=™M;8;—Ng)-
Hence 2,= }m,m,m,—(m,m,+m,+1)n{g,...(3).
For instance, let the value of the imports be originally of

the value of the currency of the country: and let the exports
1 : :

be, in the first year i of the same value; in the two successive

years let the exports be so diminished in quantity as to be 4

5
Ars . 5 s
and 6 if estimated at the same price: therefore Sp,e, = ; £; Sp,ee

1 : 1
= 58 Spies = GS

1 5 6
Here =F, M,=7>, m=z

> i

> n=

Principles of Political Economy and Taxation. 193

Therefore by (3)
aa-(5+3*1)~ tee ie = ee.
Hence the quantity of money in the country is increased in the
ratio 45: 52, and therefore, by Post. II, the prices of all domestic
commodities are increased in the same ratio. The prices of im-
ported commodities are not affected by this change, for the
foreigners will be content with a remunerating money price, and
will be prevented by their own competition from obtaining more.
By last article, equation (2),
p, Se, = Sis =F ae = aes = oe nearly.
The value of the ultimate exports and imports, at the original

: 1 ee
prices, would be nearly 7 and 4 of the original currency, and

in the equilibrium of exports and imports these values become
equal, by means of the increased quantity of gold in the country.
To find the gold in the country at the end of each of the three
years, we have
13 17 52
82 qgs"3 &3= 158? 8&1 45°)?
Also for the value of the exports in each year,
1 1

é 13
Se, pi = 783 Seap. =F SP.ee= Goes Sesps= © Spre, =a,

If we suppose the skill of the workmen of this country to
imcrease, so that the commodities which we can manufacture
cheaper than foreigners become more numerous, the value of the
exports (restrictions being supposed absent) will increase. In that
case such an importation of gold, and consequent increase of all
prices here, will occur, as has been supposed in the last and
present article.

Vol. IV. Part I. Bs

194 Prorressor WHEWELL on the

THE RATE OF EXCHANGE.

17. The different tendency of the money metals to flow into
or out of different countries, and the different level of prices con-
sequent on this, is not at all indicated by the rate of exchange.
On the contrary when the equilibrium is attained which such
tendencies produce, the exchange is at par. The average rate of
exchange can only be for or against a country when the
value of the exports is greater or less than that of the imports.
In that case the following appear to be the principles upon which,
in accordance with the views commonly promulgated on this
subject, the amount of the rate must be determined.

POSTULATES CONCERNING EXCHANGE.

I. When between two countries 4 and B, the debts due
from A to B are of greater money amount that those due from
B to A, bills in A payable in B will be sold for more than
their nominal value.

The seller in A of a bill on B, authorizes a person coming
from A to make a demand on his correspondent in B. If his cor-
respondent cannot answer this demand by a bill on A, he must
pay in coin; and if there be not, arising from mercantile trans-
actions, a sufficient number of bills on 4 to meet such demands,
gold must be exported from 4 to B for that purpose. The seller
of the bill in 4 will not therefore sell it without providing for
the foreseen expense of this transfer of gold, and the price of
the bill will rise above its nominal value, by some premium
dependent on the amount of such expense supposed to be
necessary.

Principles of Political Economy and Taxation. 195

II. The aggregate amount of all the premiums of bills on
B sold in A (supposing them to have the average value) will be
the expense of the transfer of so much gold as is necessary to
restore the equilibrium.

If the amount of the premiums were greater, it would be pos-
sible to make money by drawing bills on B, selling them, and
transferring gold from 4 to meet them. If the amount of the
premiums were less, the holders of bills would refuse to sell,
and those who had debts to discharge in B would have to incur
the expense of transferring gold: and, rather than do this, they
would pay a larger premium.

III. The premium on all good bills on B sold in 4 will be
the same.

Supposing the market of bills regulated by competition this
will obviously be true: for one good bill is on the same footing
as another.

18. I now proceed to calculate the rate of exchange under
given conditions of the imports and exports.

I shall suppose that the bills on B negociated in A are drawn
for the payment of the exports from A to B. Let Sep be the
exports, and Sis the imports, in this case. Then Sep is the
amount of the bills drawn, and Sep— Sits of the gold which it
will be necessary to transfer, in order to balance the account.
Let x be the premium on 1. Therefore «Sep is the premium
on the whole mass of bills. Also, let y be the expense of trans-
ferrmg a unit of gold from B to 4. Therefore y(Sep—Sis) is the
expense of transferring the balance. Hence, by postulates IT
and III of last article, we have

«Sep =y(Sep— Sis) ;
BBQ

196 PRoEESsOR WHEWELL on the

_ _ Sep—Sis
' * ee

whence « the rate of exchange is known.

19. In the case assumed in Art. 6, we may now calculate the
rate of exchange during the progress of the alteration.

Let x, x, x, be the rates of exchange during the 3 first years,
8 Me aan
Se, p, oir 3h7— 34

x. =)1- ais ! =}1- Ft waht
oy Se.p,s7 3957137

epee! =} - 7h os
=U) Sept Ise at!

x=}1-

If the expense of transferring gold be 1 per cent., the exchange

1 1

will be, in the Ist, 2nd, 3rd year respectively, _ wage? 'BER

, and

afterwards 0.

20. I will not quit the subject without again stating that I
by no means wish to attribute any mathematical certainty, or any
extraordinary value, to the preceding results, considered with re-
ference to their application to the real circumstances of. human
affairs. I must however observe, that if they are useless and in-
applicable, the fault resides in the postulates which I have bor-
rowed from Mr. Ricardo and others, and not in the mode of
deducing the consequences of these principles. If these postulates

Principles of Political Economy and Taxation. 197

were universally and strictly true, the results, as above stated,
would be exactly, and in all cases, verified: I may add, that if
we had reached such a point in the progress of this science—if
it were reduced to a few certain principles, and to a long train
of deductions from these—(a form which it appears to have been
the object of Mr. Ricardo and his followers to give to it—) the
mathematical method would be the one proper for its treatment,
being the most certain, the shortest, and, with a little prepara-
tion, the simplest. Mathematics is the logic of quantity, and will
necessarily, sooner or later, become the instrument of all sciences
where quantity is the subject treated, and deductive reasoning the
process employed.

I am however well aware, that the pretensions of Political
Economy to such a scientific character, are as yet entirely inca-
pable of being supported. Any attempt to make this subject at
present a branch of Mathematics, could only lead to a neglect
or perversion of facts, and to a course of trifling speculations,
barren distinctions, and useless logomachies. “ Collocatio ejus
inter mathematica” as Bacon says of another science, ‘hune
ipsum defectum et alios similes peperit; quia a phenomenis prae-
mature discessum est.” And these defects may be incurred, even
though common verbal reasoning be substituted for mathematics,
if the course adopted be that of assuming principles and defini-
tions, and making these the origin of a system. The most pro-
fitable and philosophical speculations of Political Economy are
however of a different kind: they are those which are employed
not in reasoning from principles, but to them: in extracting from
a wide and patient survey of facts the laws according to which
circumstances and conditions determine the progress of wealth,
and the fortunes of men. Such laws will necessarily at first, and

198 Proressor WHEWELL on the Principles, &c.

probably always, be too limited and too dependent on moral and
social elements, to become the basis of mathematical calculation :
and I am perfectly ready to admit, that the discovery of such
laws, and the investigation of their consequences, is an employ-
ment of far higher philosophical dignity and importance than any
office to which the Mathematician can aspire.

W. WHEWELL.

TRINITY COLLEGE,
May 7, 1831.

ERRATA.
Page Line
158 9 for extend read obscure.
173 7 from bottom, for effected read affected.
175 4 ————, for greatest read least.
7 for less read greater.
for less read greater.

|
neo

for exports read imports.

Addition to a Paper “On the Nature of the Light in
the Two Rays produced by the Double Refraction

of Quartz.”

by G. BrAtRyY, MCAS MeG:S:;

LATE FELLOW OF TRINITY COLLEGE; PLUMIAN PROFESSOR OF ASTRONOMY AND
EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE; AND FELLOW OF THE
CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read April 18, 1831.]

Arrer I had made the experiments described in a paper which
was read to this Society on February 11, 1831, I received from
Mr. Dollond an apparatus constructed under my direction, which
for convenience and extent of application exceeds any other that
I have seen. The parallel rays that fall on a piece of plate glass
blackened at the back are reflected, all completely polarized, and
are received on the first lens, which makes them all pass through
one point: then diverging they are received on the second lens,
‘whose distance from that point is equal to its focal length, and
they emerge from it parallel. In this state they are received on
the analyzing plate (a piece of plate glass blackened behind) and
all are of course completely analyzed. They are then received
on the third lens (fixed in a sliding tube, like the eyeglass of a
telescope) which makes them all pass through the eyehole. The
lenses are of equal focal length, and their arrangement is pre-
cisely the same as that of the lenses in the old three-glass eye-

200 Proressor Airy on the

piece, measuring the distance between the second and third lenses
along the incident and reflected ray. The analyzing plate, with
the second and third lenses, turns on a spindle parallel to the
rays polarized by the first plate, whose direction passes through
the centers of the first and second lenses. To see the rings &c.
produced by a crystal, it should be placed at the point where the
rays cross between the first and second lens: and a specimen
= inch broad so placed will shew the rings with perfect brilliancy
and clearness in their utmost extent. If it is wished to use a
micrometer, the micrometer should be placed between the polari-
zing plate and the first lens, at a distance from the latter equal
to its focal length, where it will be seen distinctly at the same
time that the rings are seen distinctly. If it is wished to see the
macled structure of quartz, amethyst, topaz, &c. the crystal is to
be placed in the situation assigned for the micrometer: then no
rings will be seen, but on turning the analyzing apparatus round
its spindle the different parts will be differently coloured. For
the use of plane-polarized light only, there is no need for a po-
larizing plate larger than the projection of the first lens: and the
distance between the nearest edge of the polarizing plate and the
first lens needs not to exceed by a large quantity the focal length
of the latter. But as I now consider every apparatus incomplete
which does not allow of the use of circularly and elliptically —
polarized light, I have had the polarizing plate separated so far
from the lens that it allows Fresnel’s rhomb (mounted as in
Plate 8, fig. 4) to be placed between them, room being left for
the micrometer &c. between the rhomb and the lens: and the
polarizing plate is made so large that it will transmit plane-
polarized light to the end of the rhomb, at whatever angle it is
placed, while the other end is centrally opposite to the first lens.

Double Refraction of Quartz. 201

To use artificial light with facility, a lamp is placed in the
focus of a lens whose diameter is equal to the diameter of the
circle described by the point of the rhomb. The board which
carries the lamp and lens is cut at such an angle that on apply-
ing its sloped end to the side of the board carrying the other
apparatus, the light is reflected by the polarizing plate in the
proper direction without any farther adjustment.

The advantages of this apparatus are, that with a very small
specimen, by day or by night, it will exhibit all the phanomena
of plane and elliptically polarized light to the greatest angular
extent that the eye can receive, and with conveniences of mea-
surement that have never before been given: that thus a macled
crystal, or a piece of unannealed glass, &c. can be as it were
dissected by successive examination of different small portions:
that any accidental roughness or inequality of the crystal pro-
duces no sensible effect: and that by placing the specimen in
another position, the macled structure can be exhibited with
singular clearness.

In the examination of the phenomena which quartz presents
when exposed to circularly polarized light, detailed in my paper
before alluded to, I found very great difficulty in consequence
of the contraction of the field of view when Fresnel’s rhomb is
used with the common polarizing apparatus. In examining the
two spirals inwrapping each other, I could see little more than
a single line at a time: and it was only by carefully turning the
crystal that I could discover the relation of one line to another.
It was in fact principally to overcome this difficulty that I devised
the apparatus here described: with this I can at once see the
folds of the spirals as far as the colours are sensible. I have
much satisfaction in seeing that my delineation was quite correct.

Vol. IV. Part I. Cc

202 Proressor Airy on the

On considering my hypotheses relative to the nature of the
two rays of quartz, the following method suggested itself as a
means of verifying one part of the hypotheses, and as affording
a power of measuring the ellipticity of the rays. Suppose (by
placing Fresnel’s rhomb in a position between 0° and 45°, or
between 90° and 135°) elliptically polarized light is made to pass
through quartz. Whether this be right-handed or left-handed,
there is one direction (4) in which one ray of the quartz (sup-
pose for instance the ordinary) is of just the same kind as the
incident elliptical light. Consequently that light furnishes no
extraordinary ray. Now if we take a direction (a) nearer to the
axis by the smallest possible angle, and another (4) further from
the axis by the smallest possible angle, than the direction just
mentioned (4), the same elliptical light incident in the directions
(a) and (4) will furnish extraordinary rays, but the paths of these
extraordinary rays will differ (independently of all other causes) by
half the length of a wave. For the elliptical light which is of
the same kind as the ordinary ray in (4) is more elliptical than
the ordinary ray in (a) and less so than that in (4). And there-
fore when we separate: the elliptical light into an ordinary and
an extraordinary ray in (a), it is the defect of its minor axis
which produces the extraordinary ray: when we do the same
for (6), it is the excess of its minor axis which produces the
extraordinary ray. The vibration therefore which produces the
extraordinary ray in (a) bemg in the positive direction, that which
produces the extraordinary ray in (4) will be in the negative
direction, or vice versa. And this amounts to the same as re-
tardation or acceleration by half the length of a wave. It will
readily be seen that this is independent of the crystalline separa-
tion of the two: rays, and is true however small be the angle

Double Refraction of Quartz. 203

between the directions (a) and (4), provided that one be nearer to
the axis, and the other further from it, than (4). Now it is well
known that the order of the rings depends on the number of
lengths of wave which the extraordinary ray is in advance or
retardation of the ordinary ray. At this place the advance or
retardation is suddenly altered by half a wave. Consequently the
order of the rings is suddenly altered by half an order: the rings
become faint, and then there is a saltus of half an order in the
colours. And the direction of the ray where this saltus takes
place being observed, it gives the direction in which the ordinary
ray has the same ellipticity as the incident light, which is known
from the position of the rhomb.

By this reasoning I had satisfied myself that the relation be-
tween the direction of the ray and its ellipticity could be made
evident to the eye. On trying it with the apparatus just de-
scribed, I found that the appearance was exactly what I expected.
On placing the rhomb in position 315°, the spirals are perfect.
On turning it forwards, the internal folds break successively, (if
the crystal be left-handed) in a line nearly horizontal, and the
upper part of each unites itself with the lower part of the fold
next beyond it. This continues, the outer folds being broken
after the inner ones, till when the rhomb has reached 0° the ap-
pearance is that of perfect circles. On turning still forwards, the
successive circles break (beginning with the outer ones) in a line
nearly vertical: and when the rhomb has reached the position
45°, the spirals are perfect as before, but twisted 90°. Thus at
any position of the rhomb intermediate to 0°, 45°, &e. one or
more of the inner circles are complete, but distorted, and the ex-
terior circles are changed to two inwrapping spirals. It is plain
that the points where the spirals begin are the points where the

cca

204 Proressor Airy on the

elliptical light supplies only the ordinary ray, as the colours one
quarter of an order within these points, and one quarter of an
order beyond them, are the same. _

The conclusion, that the inner rings will be circles, and the
outer ones two spirals, follows easily from this consideration.
The intromitted elliptical light has a smaller minor axis than the
ordinary ray (which is the only one that it resembles) in the
directions nearest to the axis of the crystal, and therefore it will
produce curves analogous to those produced by plane-polarized.
light; that is, analogous to circles. But it has a Jarger minor
axis than the ordinary ray, in the directions far from the axis of
the crystal, and therefore it will produce curves analogous to
those produced by circularly-polarized light; that is, analogous
to two spirals infolding each other. There is no difficulty in
making a more accurate investigation, on the principles described
in my former Paper: but I have not done it for a reason that
will shortly appear.

I have not yet had the opportunity of making measures which
are sufficient to point out the law that connects the ellipticity of
the rays with the angle that they make with the axis. The fol-
lowing points however are made out. One of the rays certainly
is right-handed elliptical, and the other certainly left-handed
elliptical (or so nearly, that no difference is distinguishable).
The major axis of one is certainly perpendicular to the principal
plane of the crystal, and the major axis of the other is certainly
in that plane.

In some trials of measuring the ellipticities of the rays, I
seem to have arrived at the following conclusion. The propor-
tion of the axes of the ordinary ray is more nearly one of
equality than the proportion of the axes of the extraordinary

Double Refraction of Quartz. 205

ray. For instance, with a right-handed plate of thickness 0,38
inch, and using a red glass, the first red ring was rendered am-
biguous (if I may use that term to denote the state when the
ring is broken in such a manner, that it is difficult to say
whether the part on one side is most nearly connected with the
exterior or interior part on the other side) by supplying an or-
dinary ray only, with elliptical light, whose

minor axis

: —— = tan 17°. 15’;
major axis

er by supplying an extraordinary ray only with light, whose

minor axis
ee AUT Bes

: — = tan 16°. 2’.
major axis

With a left-handed plate of thickness 0,16 inch, the first red ring
was rendered ambiguous by supplying an ordinary ray only with
elliptical light, whose

minor axis _

: —- = tan 9°. 3’,
major axis

or by supplying an extraordinary ray only with light whose

minor axis

= — = tan 8°, 50’.
major axis

The first result is the mean of 8 measures of each, and the
second the mean of 4 measures*, The zero points of the rhomb-
graduation were determined by observing when the rings of cale
Spar were not broken. This determination is very accurate; but
if it were faulty, the effect of any error in the zero point, as
well as of any imperfection in the construction of the rhomb

* In the first of these cases, (by a rough measure), the rays made with the axis.
of the crystal, the angle 9°. 42° in air: and in the latter 13°. 50' in air.

206 Proressor Airy on the

(such as I believe to exist in it) would operate in different ways
with right-handed and left-handed plates: and it is on this ac-
count that I have given both. No error is to be apprehended
from the jogging of the circle carrying the rhomb, as it is pro-
vided with four verniers, at 90° apart, all which were read. If
this point should be made out, it may be only a consequence of
the separation of the rays within the quartz: or it may be
another anomaly to be added to the already sufficiently compli-
cated phenomena of quartz. At all events, regarding the perfect
equality of the ellipticities as doubtful, I have carried no farther
the investigations made on that supposition: though the differ-
ence of the ellipticities is so small, that their error would be
insensible.

To any one who wishes to proceed with these experiments,
the following hints may be of some use.

It is convenient to have a wire carried by the rhomb, parallel
or perpendicular to the plane of reflection within the rhomb, and
in the place which I have mentioned as the position for a mi-
crometer: as the observer will then see that a line from the
‘center of the rings to the point where the ambiguity takes place,
is either parallel or perpendicular to the wire.

The elliptically polarized light is of the same kind (always
right-handed) when the position of the rhomb is between 0 and
90°, or between 180° and 270°: and of the same kind (always
left-handed) when the position of the rhomb is between 90° and
180°, or between 270° and 360°.

The proportion of the axes is the tangent of the reading of
the rhomb-position. The major axis is parallel to that edge of
the end of the rhomb which makes the greatest angle with the
plane of reflection at the polarizing plate.

Double Refraction of Quartz. 207

When the ambiguity takes place near the right or left hand
of the center, it is the ordinary ray only which is furnished.
When it takes place nearly above or below the center, it is the
extraordinary ray only which is furnished.

The following conjecture may, perhaps, without impropriety, —
be attached to this Paper. It is the suggestion of an explanation
of the unequal refrangibility of differently coloured rays.

To account for the difference of refrangibility, we must sup-
pose that the velocity of waves of different lengths is different
either in air, or in the refracting medium, or in both. If it were
ditterent in air, it would affect the aberration of stars by a quan-
tity that might be sensible: there is no reason to think that this
is true. It is probable therefore that the difference is wholly
within the refracting medium. Now it is particularly to be re-
marked, that the difference of velocity does not depend on the
magnitude of vibration of each particle, for it is the same,
whether the light be feeble or intense, that is, whether the vi-
bration be small or great. Nor does it depend on the relative
vibration of two contiguous particles, as that varies in the same
proportion as the last, with a variation of the intensity. The
only element which, in conjunction with either of these, will
define the undulation, is the time of vibration: and it is in fact
the time of vibration which distinguishes the different kinds of
light. It would seem natural therefore to seek for an explanation
of the difference of velocities in something which depends not
on space, but on time. Now we have every reason to think
that a part of the velocity of sound depends on this circumstance :

208 Proressor Airy on the Double Refraction of Quartz.

that from the suddenness of the condensation of the air, the heat
evolved by that condensation has not time to escape, and the
elasticity is therefore greater than if it had been slowly con-
densed: that, in fact, the law of elasticity is altered. Now the
conjecture which I have to offer is, that perhaps there may be
in refracting media something depending on time which alters
their elasticity, in the same manner in which heat alters the
elasticity of the air: that as in air the elasticity is greater with
a quick vibration of particles than it would be if the vibration
were exceedingly slow, so also in the refracting media, the elas-
ticity may be greater with a quick vibration, than with one some-
what slower. Or perhaps the contrary effect may follow: if the
vibration be quick, the latent heat (or whatever it is) may not
have time to come to the exercise of its influence on the elas-
ticity. In the latter case the elasticity, and consequently the ve-
locity of transmission, would be greatest for the slowest vibra-
tions (that is for the red rays) and therefore they would be the
least refracted. I am not prepared to say whether the general

law of superposition of small vibrations would hold on this sup-
position.

G.B. AIRY.

Observatory,
April 13, 1831.

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VI. Description of Chiasognathus Grantii, a new
Lucanideous Insect forming the type of an unde-
scribed Genus, together with some brief Remarks
upon its Structure and Affinities. In a Letter ad-
dressed to one of the Secretaries.

By J. F. STEPHENS, Esq. F.L.S.
[Read May 16, 1881.]

My Dear HEeEnstow,

Tue magnificent Beetle submitted by you to my
inspection proves to be, as I anticipated, not only perfectly novel
to science as a species, but forms the type of a genus as in-
teresting from its structure, as it is remarkable for its splendour
and colourig. I shall therefore very briefly notice some of its
peculiarities, though I cannot but regret that from the circum-
stance of my attention having been chiefly directed to indigenous
entomology, the task should have devolved upon one so little
conversant with exotic forms, and more especially as Great
Britain, is remarkably deficient in the group to which the present
insect belongs.

The Lucanide, to which family Chiasognathus appertains,
are distinguished amongst other characters by the extraordinary
developement of the mandibles in the males, which in the
common Stag-beetle rarely exceed half the length of the body,

Vol. LV. Part II. Do

210 Mr. STEPHENS’s Description

but in Chiasognathus they acquire an elongation exceeding that
of the body; they are extremely strong and robust at the base, and
evidently capable of biting very sharply; towards the middle they
become flattened, and at the tips they are incurved so as to cross
over each other—whence the origin of the name I have applied
to the genus*—the internal edge-is irregularly serrated throughout,
with a large tooth towards the base, and the apex has an acute
recurved hook (Plate X. Fig. 4). In the genus Lucanus a small
tubercle may be observed at the outer base of each mandible;
in the insect now under examination this is greatly developed,
and forms an acute spinous process about one-third the length
of the mandibles, serrated within, and pointing inwards, so that
in situ the two cross each other towards the apex similarly to the
mandibles themselves.

The upper lip (labrum,) is very distinct, being composed of
a coriaceous plate with a strong rib down the centre.

The lower jaws (maxille, Plate IX. Fig. 4.) are small, but the
apical portion is very long and delicate, and fringed with very
slender hairs:—the maxillary palpi are elongate, slender, with the
basal joint very short, the second nearly as long as the others
united, sub-clavate, the third half the length of the second, the
terminal elongate, somewhat attenuated.

The lower lip (labium, Pl. 1X. Fig. 8.) is membranaceous, with
two, rather broad, flat, elongate, lacinize, the extreme edge of which
is finely ciliated :—the labial palpi are short, with the basal joint
very short, the second rather longer, the terminal one nearly as
long as the other two united, and attenuated.

* XiaCw decusso, Tvaboc maxilla.

of Chiasognathus Grantii. 211

The mentum (P1. IX. Figs. 5. and 7.) is transverse, semicircular,
and notched anteriorly.

The Antenné are remarkable for the extraordinary elongation
of the basal joint and the whorl of hairs which ornament its tip; the
remaining joints are short, the three first somewhat obconic, the
two following transverse and produced within, the four terminal
ones also produced within; the process being longer, and the
articulations more distinctly laminated.

The furcate anterior portion of the head (clypeus), and above
all, the distinct existence of four eyes, as well as the great
strength of the fore legs, are characters of no little importance :
to which may be added the superb colours with which nearly
the entire msect is adorned; the castaneous golden-bronze of the
elytra, the burnished golden-green of the gibbous centre of the
thorax, and the iridescent hues of its sides and of its posterior
spines, form an assemblage of intense tints rarely united into one
form. In fact every part of this unique insect possesses characters
of extreme interest, as may be clearly perceived through the
medium of the accompanying figures, executed by my friend
Mr. Westwood, who in his delineations observed several peculi-
arities which he kindly pointed out to me.

The food of the Lucanide consists of the flowing sap of de-
caying trees, which in the typical genus is lapped up by the four
plates or laciniz of the maxille and lower lip; but in this insect
the very arched form of the mandibles appears to form an ob-
stacle to the application of the laminzw to the tree unless the
mandibles be opened to a great extent, as may be readily seen by
the lateral view (PI. X. Fig. 3.) given in the accompanying figure.

Respecting the affinities of this insect, the genus which makes
the nearest approach to it is evidently Pholidotus, with which it

DD 2

212 Mr. STEPHENS’s Description

somewhat agrees in the structure of its maxillz and of the in-
ferior portion of the trophi, excepting that the mentum is gla-
brous; the genus above mentioned is manifestly allied to Lam-
prima, and these two genera with Chiasognathus contain the only
Lucanideous insects (with the exception of Platycerus) that are
adorned with metallic colours. In Lamprima the maxille are
short; the basal joint of the antenne shorter than the remainder
taken together; the mandibles slightly elongated; but in Pho-
lidotus the maxilla are elongated, being furnished with a peni-
cilliform process as in Lucanus, the three apical joints of the
clava of the antennz are alone enlarged, and the basal joint is
longer than the remainder, the mandibles are large, clothed with
down on their inner surface, and the mesosternum is slightly pro-
duced in front as in Lamprima. Chiasognathus therefore, by
varying from the above allied genera in several of these parti-
culars, makes a near approach to Lucanus, which is the only
genus of the group containing species which may vie with it in
bulk, strength of mandibles, habit, and general conformation ;
hence it evidently forms a truly beautiful and interesting link
between the two conterminous genera Lucanus and Pholidotus,
or the two families Lucanide and Lamprimide, possessing the
gigantic structure of the former, and the resplendent hues of the
latter family.

of Chiasognathus Grantit. 213

CHIASOGNATHUS.

Antenne fractz, articulo primo longissimo, sub-flexuoso, gracili, ad
apicem incrassato et fasciculo pilorum instructo, clava pectinata.

sex-lamellata, articulis quatuor ultimis sub-zqualibus.
Labrum distinetum, sub-coriaceum, carinatum.

Mandibule \ongissime, intus serrate, apice incurve et decussate, basi

incrassate, subtus processu spiniformi, intus serrato, armate.

Mazxille processu apicali longissimo, exserto, gracili, sub-attenuato,

pilis brevibus instructo.

Palpi mazxillares elongati, graciles, articulo primo brevissimo, secundo
longissimo, sub-clavato, tertio dimidio breviore, ultimo elongato.

sub-attenuato.
Labium membranaceum, laciniis duabus, sub-latis, elongatis, armatum

Palpi labiales breves, articulo primo brevi, secundo longiore, tertio

longissimo, attenuato.

Mentum transversum, semicirculare, natice emarginatum.

Corpus depressiusculum, sub-latum. Caput latum, transversum,
sub-triangulare; clypeo furcato emarginato, versus latera utrinque sub-
spinoso. Oculi quatuor, convexi. Thorax truncato-conicus, antice in
medio rotundatus et pone oculos excisus, lateribus dilatatis, deflexis, ad
angulum posticum profundé emarginatis, spinis duabus acutis ar-
matis, basi bisinuato. Scutellum rotundatum. Pedes elongati, femoribus
anticis magnis, sub-cylindricis, sub-glabris, intermediis et posticis graci-
lioribus, tenué pilosis, marginibus anterioribus et posterioribus dense

214 Mr. STeEPHENS’s Description

ciliatis; tbs anticis elongatis compressis, apice emarginato, externé
sub-convexis, interne sub-planis, versus apicem paulo curvatis, prope
marginem anticum spinis acutis armatis, extus multidentatis, dentis
duobus apicalibus magnis; intermediis et posticis brevioribus, teretibus,
apice interno bicalcarato, extus serratis; éarsi articulis quatuor basalibus
sub-zqualibus, ultimo elongato clavato, unguiculis duabus compressis,

acutis, curvatis.

Sp. 1. Grantii. Sub-viridi-aureus, cupreo refulgens, thorace gibbo viridi
intense, lateribus angulisque jposticis versicoloribus, elytris sub-
castaneo-tinctis, femoribus aureo-viridibus, tibiis cupreo-ferrugineis,
tarsis antennisque nigris, abdominis segmentorum marginibus tes-
taceis, pectore lanugine griseo tecto. (Longitudo corporis
3 3 ine. 34 lin).

In honorem Di. Geo. Grant, M.D. hoc splendidissimum insectum

nominavi.

DESCRIPTION.

Mandibles (of the male) rather more than half the length of
the body, finely but distantly punctured, with a few longitudinal
wrinkles at the base, the inferior process very glossy and im-
punctate, colour rich copper, tinted with brilliant shades of a
golden hue, the base and its appendage rich’ blue-green and
iridescent; the curvatures bronzed-black; head bright golden-
green, varying in tint with the light, the disc very glossy, re-
motely, but finely, punctate, bluish; the lateral and posterior
margins of a variable golden hue, with the spinous processes of

of Chiasognathus Grantii. 215

the clypeus purplish bronze: eyes glaucous: thorax unequal,
truncate-conic, with the anterior margin rounded in the middle.
and gradually excised on each side behind the eyes, the anterior
‘lateral angle acute; the lateral margins dilated and deflexed.
with a deep impunctate fovea on each, and the hinder angle
with a profound circular excision terminating in two spines, of
which the posterior one is longest; the base is laminated; the
disc gibbous, rugose-punctate anteriorly, very glossy and impunctate
posteriorly, and produced into an abbreviated ridge, composed of
two lunules; the colour is intense golden-green, with the ante-
rior and posterior margins rich purplish-copper, varying in cer-
tain positions of light to blue and violet, especially towards the
posterior angles, and the fovee on the lateral margin are fine
purplish-green ; seutellum of this last tint, impunctate, its base
clothed with short pale hairs in the centre ; elytra very finely gra-
nulated throughout, except the humeral elevation, which is
smooth, of a greenish-chesnut, with a coppery or golden hue, the
suture, apex, and a narrow indeterminate streak towards the
lateral margin somewhat ferruginous, the lateral margin itself and
the reflexed portion of the el ytra bright green with iridescent tinges.
Beneath: the sides of the head are bright green, slightly iride-
scent, and regulose-punctate, the palpi and throat black, the
latter impunctate, the mentwn deep violet; breast purplish-
copper, slightly punctate on the sides, with a few transverse
wrinkles between the anterior legs: abdomen, anteriorly, or
rather post-pectus, densely clothed with pale lutescent hairs, the
remaining portion bluish-green, slightly pilose, with the margins
of the segments pale testaceous: femora finely pubescent above.
glabrous beneath, the four hinder ones producing a dense fringe
of pale hairs anteriorly and posteriorly, all of a bright bluish-

216 Mr. Srepuens’s Description of Chiasognathus Grantii.

green, tinted with coppery above; ¢ébie above coppery-green,
with a tendency to castaneous, beneath rusty-chesnut, tinged in
certain directions of light with greenish; the serrations, tubercular
processes, and spines at the apex purplish-black; tarst brown-
black; antenne the same; with the singular fascicle of hair at
the apex of the basal joint pale griseous. At the base of the
mandibles exteriorly, and on the sides of the thorax are a few
very short pale scattered hairs, and on the ridge which divides
the eyes is a long delicate fringe of similarly coloured ones;
the anterior and posterior margins of the thorax both above and
below are also densely ciliated with short pale hairs.

The female is unknown, but in all probability, when disco-
vered, the mandibles will be found to be abbreviated, and the
form of the clypeus and thorax slightly different from those of
the male.

IT am, Your’s, &c.

J. F. STEPHENS.

P.S. Dr. Grant, the gentleman who presented this interesting spe-
cimen to the Society, was surgeon on board H.M.§. Forte, when she
returned to England in the summer of 1830, from the South American
station. The Insect was brought to him in January by a native, who
stated that he had found it on a resinous shrubby plant in the Island
of Chiloe, which is separated from the Main Land at Valparaiso by a
very narrow channel,—It appeared to have been a recent capture.

217

EXPLANATION OF THE PLATES.

Plate Fig.

ID: ile po
Cutasocnatuus Granti, under side!

i Mazille with lacinia and palpus.
4,

5. Mentum, processes of labium and palpi, under view.
6. Base of anterior femora.
7. Mentum, labium, &c. upper view.

8. Labiwm with processes and palpi, lateral view.

X. 1. CurtasocNatHus GrantTII, upper view.
2. Mandibles, head, thoraa, &c.
3. Lateral view of ditto.
4. Apex of mandibles.
5. Antenne.

6. Tarsus.

Vol. IV. Part IL. Er

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VII. A Case of Hanan Monstrosity, with a Commentary.

By W. CLARK, M.D. &c. &c.

LATE FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF ANATOMY IN THE
UNIVERSITY OF CAMBRIDGE.

[Read May 16, 1831.]

Or late years no subject has more incessantly occupied the
labours of learned continental Anatomists, than the investigation
of the steps by which the rudimentary organs of embryos advance
to their perfect form. Nor has any proved more fertile in results
the most valuable. It was observed that in all vertebral ani-
mals at least, that process is effected according to a plan which
is uniform for all: that in the lower orders, the full development
of this plan is arrested in its progress; whilst in the higher, ac-
cording to their place in the scale, it is constantly advanced to
a still increasing degree of perfection. By a skilful application of
this general principle it was, that Cuvier and Geoffroy St. Hilaire
were so successful in explaining the osseous system in the crania
of reptiles, fishes, and birds—which, up to thei day, had been
nearly unintelligible. And thus also, that Serres and Tiedemann,
and Gall and Spurzheim, succeeded in assigning their true import
to the different portions of the brain in birds and in fishes, by
shewing to what parts of the rudimentary brain of the higher
class they are analagous. The nature of such enquiries necessarily

EEQ

220 Proressor Ciark on a Case

drew the attention of those engaged in them, to the consideration
of those unusual or imperfect forms of animals which had hitherto
been considered to defy subservience to general laws, and which
had been collected in a heterogeneous mass, as objects of igno-
rant curiosity, or as proofs that nature is sometimes capricious,
and disdains an absolute submission to those forms which she
herself had consecrated. An extended enquiry soon led to the
conclusion that all the known aberrations from usual standards
may be referred to one of three orders: according as they are
characterized by detect, or by excess in the development, er by
the inversion of parts. With respect to anomalies from defect,
it was further ascertained that those organs in which they occur
have been arrested in some one of those transient stages through
which they were passing to a higher degree of perfection: and
that in these stages they resemble the permanent condition of the
perfect organ in some lower order of animals*. The explanation
of the two other orders has not hitherto been so satisfactory.
With respect to excess of development, though it has been clearly
determined, that no organ in an individual of a lower class, how-
ever it may deviate from its perfect type, ever represents the type
of a higher class: yet, the various instances of this mode of de-
yiation have seemed so entirely to militate against general rules,
that some distinguished anatomists have considered each case as
forming a genus of itself, and as subject from its earliest period
to its own peculiar laws. May it not be, however, that in the
production of these unusual animal forms, the law appears to be
peculiar merely because its ordinary expression or effect, has
been disturbed by causes which it is very difficult to assign, and

* In a great many instances also the cause of the arrest has been assigned.

of Human Monstrosity. 221

when assigned to estimate? It is with pleasure that I embrace
an opportunity of proving, as I hope to do, that, in one instance
at least, the apparently excessive development is referable to the
usual laws.

The monstrous production which I proceed to describe, con-
sists of the junction of two males by the head, neck, sternum,
and abdomen of each, as low down as the passage for the common
umbilical chord. From this point the bodies are separate. The
external organs of generation are natural, the testes on the point
of descending into the scrotum, the opening of the rectum, perfect
in both. There are two arms, and two legs to each foetus, and
the whole body of each, from the umbilicus downward, is well
nourished, and very perfectly moulded.

The connexion is of this kind. The spine of one foetus is to
the extreme right, of the other to the extreme left. The ribs from
each spine arch towards a mid plane, and thus form a body with
breasts on each front common to the two. The occiput of each
arches from its spine, upwards, backwards, and then forwards,
until it meets the corresponding parts from the other spine. And
then a mixing together of the two heads takes place. The faces,
in this operation are not destroyed, for the right portion of the
face of one foetus joins with the left portion of the face of the
other, to make up two anomalous countenances, which look
forward and backward, Janus-like, in a direction perpendicular
to the common plane of the spines.

One of these faces is more perfect than the other. There are
two ears to each, and a mouth: over the mouth is, In each, an
eye-ball, surrounded by four eye-lids, of which the external angles
are naturally united, and the internal angles meet in the mid line
of the face, above and below the eye-ball. In the more perfect

222 Proressor CLarkK on a Case

face, the eye-ball presents two cornez of different sizes: in the
less perfect, one large transparent disc. Plate 11.

In the less perfect face, below the junction of the eye-brows
is a very small fleshy tubercle, which is the only rudiment of
a nose. In the other face this tubercle is considerably larger :
is evidently bony from its resistance: is covered by a loose flap
of skin: and, when this is turned down over the eye, the tubercle
is found to present a cell on either side, separated by a middle
septum. Plate 12. Fig. E and F.

The mouth in the more perfect face is fairly formed. When
the lips are separated, a well developed tongue is seen. The
mouth of the less perfect is much smaller. No tongue is here
visible: but this, as will be afterwards found, is only an apparent
defect. This mouth opens by an aperture behind, about a line
in diameter, into another cavity: into which cavity the mouth of
the more perfect face also opens.

With respect to the osseous system, I shall detail only what is
peculiar and unusual: the bones of the spinal column, of the
pelvis and limbs of each foetus, and the ribs, being normal.

The bones of the head, standing upon each vertebral column,
unite to form a single cavity for the common brains of the two
foetuses.

On the top of either column is placed an occipital bone to
which the temporals and parietals are attached in the usual way.
The parietal bones, corresponding to one column, meet those cor-
responding to the other column, at their anterior and superior
angles which are truncated; membrane alone uniting the truncated
angles, and thus forming on the vertex of the common head a
large fontanelle. Between the truncated and the anterior inferior
angles of the corresponding opposite parietal bones, are situated the

of Human Monstrosity. 223

frontal bones: that which is above the less perfect face being much
smaller than the other, and formed of one portion only, whereas the
other, as is more common, is formed of two. Plate 12. Fig. 4, B, C, D.

It is interesting to determine in what way the bones of the
two crania are modified and connected so that two faces may be
formed in planes parallel to that of the spines. To this we
shall be led by a consideration of the common basis of the
erania. Plate 13. Fig. K and L.

The occipital, basilar, and temporal bones are naturally con-
nected on either side, and except that the squamous portions
of the latter are very small, and the petrous portions very large,
present nothing remarkable. The sphenoid is connected by its
body behind to the basilar bone, by its great ale to the anterior
edge of the petrose and squamous bones, and by its ingrassial
processes to the frontal bones in the normal condition. And these
connexions, of course, prevail here also. But from the smallness
of the squamous bone, the ale of the sphenoid are thrown into
a direction much posterior to what is their usual direction. The
same disposition occurring with respect to the sphenoid of the other
foetus, the bodies and two great alz, on opposite sides, oppose
each other; a space intervening which is nearly quadrangular.
This space is occupied by the united ingrassial processes of op-
posite sphenoids closely compressed, (and therefore united by
ossific matter) between the right great ale of one, and the left
of the other. The great ala and bodies are also united behind
these processes from pressure ; and, the bodies still maintaining
their natural connexion with the basilar bones, thus surround
a central space which is oval in form, is covered by fibrous
membrane, and is exactly above that common cavity into which
the two mouths open.

294 Proressor Criark on a Case

The «xthmoid bone is entirely wanting on the side corresponding
to the less perfect face, and the frontal bones are there united by
ossific matter to form the roof of the orbit and the forehead of this
face: the two orbits being necessarily imperfect from the absence of
the zthmoid, and therefore forming one cavity. On the side cor-
responding to the more perfect face there is a rudiment of the
zthmoid, which is, in fact the proboscis of this side. It appears to
have been forced downwards and forwards by the lateral pressure
of the frontal bones, and is connected to their inner edges by
ligament.

Thus the smallness of the squamous bone, and of the ingrassial
process, with the absence of the zthmoid, renders the position of the
frontal bone and of the face necessarily lateral in respect of the spine.

In the less perfect face, from the total absence of zthmoid
and nasal bones, and the narrowness of the frontal bone, the
superior maxillary bones are so much compressed Jaterally that
their palatine plates form a ridge, rather than a horizontal lamina.
Hence the cavity of the mouth is deep, but very narrow; so
narrow that the tongue though small cannot project into it, but
lies in that common cavity which is between the two mouths, and
directly below the oval opening of the sphenoids. The other
mouth, from the width of the frontal bones, &c. is large enough
to receive a well-formed tongue. The membrane which lines the
sides and vault of either mouth, and admits the parotid ducts,
presents a small papilla in the situation of the uvula, on whose
sides are seen the openings of the eustachian tubes. Behind the
tongues, which are properly attached to hyoid bones, and these
to the temporal bones, is the common sac which is the bag of
the pharynx. Into this the two larynges open, covered by their
epiglottides ; and it terminates in a single zsophagus.

of Human Monstrosity. 225

The structure of the walls of the thorax may be inferred from
the description of the external appearance already given. There
are twelve pairs of ribs to each foetus. The right sternal series
of one foetus, and the left of the other are received into a common
sternum: whilst the left series of the first, and right of the
second are similarly received into the other sternum.

The cavity of the thorax is divided into two portions by a
membraneous partition proceeding from either vertebral column.
In each cavity is a perfect heart in its pericardium, with a pair
of lungs: and between the two hearts pass the two trachee ; and
the single cesophagus between them. The hearts are entirely un-
connected. And the lungs have this singularity, that of the pair
attached to a heart, one is the right lung of one foetus, and the
other the left lung of the other foetus. The thorax is closed
below by a large diaphragm, with four pillars; two to the lumbar
region of each foetus, with united tendinous centres. In_ this
diaphragm there are double the number of ordinary foramina,
one excepted, since there is but one cesophagus.

The esophagus opens into a stomach (24 inches long by 13 in
width) which rather represents a square pouch than the ordinary
curvilinear form of that viscus. One extremity may be called the
cardia, because there is a spleen closely attached to it. There
is another spleen also, but it is attached to the stomach by a
much wider peritoncal fold. From the stomach proceeds the
small intestine corresponding to the duodenum; it is a wide
straight intestine doubled upon itself, like the beginning of the
cecum in ruminating animals, and lying across the upper part
of the common abdominal cavity. To this duodenum there passes
the biliary duct from a large liver with a gall bladder belonging
to ene foetus, as 4: but it is not pervious so far as the intestine.

Vol. IV. Part II. Fr

226 Proressor CLarK on a Case

There is also a pancreas. The length of this portion of intestine
is 94 inches long. It opens into a sac nearly of the same dimen-
sions and form as the stomach, situated in the common abdominal
space immediately over the umbilicus.

There is another liver belonging to the foetus B, with a gall
bladder and ducts. The common duct is perforate for a very
small space, and ends in a band of condensed cellular substance
covered by peritoneum: which band is connected, by one ex-
tremity with the second spleen, and by the other with the lower
portion of the duodenum already mentioned. To the sides of the
sac, which filled the lower part of the common abdomen above
the umbilicus, are attached two coils of small intestine, each
closely packed, and each opening into, or arising from, the sac.
The coil, peculiar to the foetus 4 is 143 inches long, that to the
foetus B 11 inches. They both terminate in a ceecum with an
appendix vermiformis 14 long in each. The large intestine from
the coecum to the anus, is, for the feetus 4 14 inches long, for
B 15 inches. The large intestines in both have this peculiarity,
that they are attached to a fan-like mesentery, and form coils
like the small intestines in the natural state. Their proportional
length is unusually great. Plate 13. Fig. J.

The urinary organs in either foetus are altogether natural.

The common umbilical chord consists of four arteries, and two
veins; a vein to each liver, and the arteries as usual.

The arteral and yenous systems are similar for each heart and
each foetus.

From the left ventricle arises an aorta which, instead of the
three trunks which usually spring from its arch, sends off only
one: and this divides almost immediately into the common carotids
for one face, or (more correctly) into the right common carotid of

of Human Monstrosity. 227

one foetus and the left of the other. These arteries present nearly
the usual subdivisions, the right carotid is the largest, has a much
more oblique course to the head than the other: it also gives off
the greater number of branches, the arteries of the thyroid in a
great measure coming from it alone. It divides as usual at the
angle of the jaw. The other carotid, also sends off a thyroid, facial,
imternal maxillary, occipital, temporal, &c. but these are small
branches, and it principally goes to the brain as internal carotid.

The aorta then arches gently downwards to the spine of the
foetus B. It then gives off the left subclavian to this foetus next
the right, and from these the two vertebral arteries, &c. arise. It
finally terminates in the two umbilical arteries of this foetus after
supplying all the parts nearly in the usual manner.

The veins which return the blood from these arteries have a
curious destination. The ascending cava enters the right auricle
of the other heart, and the azygos and right subclavian veins
join the descending cava of that heart also: whilst the veins ac-
companying the arteries which arise from the arch of the aorta,
and the left subclavian vein form the descending cava of the
same heart from whence those arteries arose.

The arterial and venous systems of the other heart are exactly
similar to what has been described. So that the brain of one
foetus derives its internal carotid on one side and its two basilar
arteries from one heart, and the other internal carotid from the
other heart. It follows that the faces and eyes, moulded though
they be from corresponding parts of different foetuses, derive their
blood, nevertheless, from the same heart.

The dura mater passing from the occiput and tentorium of one
foetus in the usual way, joins that coming similarly from the other,
to form a common falx. The cerebral lobes are moulded into

one mass on either side of the falx, and present no appearance
FFQ2

228 Proressor Crark on a Case

of convolutions. Neither is there any sulcus on the upper surface
of that side of the brains which corresponds to the less perfect
face. On the corresponding surface of the other, however, is a
deep oblique fissure, as though the anterior lobes of these oppos-
ing hemispheres had not united throughout. The ventricles on
either side form a common cavity.

The crura cerebri of the opposite brains are melted together
over the oval aperture of the united sphenoid bones. From this
point the tubera annularia diverge separately: and are connected
to well formed, distinet cerebella. The medullze oblongate and
spinales proceed from them naturally.

There is no vestige of any olfactory nerve. The optic nerves
are very small. The right optic tractus, of one feetus, converges
to its ingrassial bone; at which point it meets the left optic tractus
of the other foetus. Here the two unite to form a single optic
nerve for either face.

There is nothing remarkable in the other nerves. They have
their usual apparent origins, and their usual passages through the
cranial bones. The left vagi send a recurrent nerve round the
aortas: the right do not surround any artery. They appear to
waste themselves almost entirely in forming the pulmonary and
cardiac plexus, and thus connect the separate sympathetic systems
in the neck and thorax.

Dimensions supposing the more perfect face to look forward:

Right Fetus. Left Fetus.
Total length from vertex to heel..... 14.5 inches...............13-9 inches.
From vertex to common umbilicus... SOARS CODE LEDER ote a NinaaEnee
Shoulder point to shoulder point..... Bion tedcctecsecnteonassiees ah Deedee.
Shoulder point to end of fingers.,.... 620)... G50 Geisesks
Raurid therel po weecnacbaces Sacaadvesn oe 1D. beens cn nea tema ce pecinke: papa) saeee ops
Length of foot ....escecseeververseeses 220 sevneesees Me iossans -

of Human Monstrosity. 229

Circumference of common head................. 12 inches.
SR Sentara dan Bee cCH aor e ee NechaA5i.. 2a Tos cmwe tf coeet es
SralProinnisicie[ecicte ba oieicteitia soe thorax) 270 sascunasenlle pees

Diameter from occiput; to oecipalsre.s...csssee 5 nO saan
wear ccccvcccccce EYE LOlCV Cre a vanisleoniseniesancnieciny Aisl0, Jone

Such being the structure of this curious production, we are
naturally led to admire those deviations from the usual disposition
of essential parts which seem to fit it for an independent existence
after birth. The respiratory, the circulating, the nutrient, the se-
cretory apparatus are perfect, and that it did not live, is to be
attributed to accidental causes, which are unknown. Here also
we have an instance of one creature, admirably compacted from
the parts of two. For the union of the instruments of intelligence,
viz. the cerebral lobes, and the nerves which supply the organs
of sense, constitute it one individual. The spinal chords also are
intimately united, since the corresponding crura cerebri to which
they pass, form one and the same mass. The curious disposition
of the lungs, and the connection of the arterial and venous system
of either body with different hearts, are evidently causes which
tend to the same effect. For without this, or something equivalent,
the circulation which is now one for the whole mass, would have
been ¢wo, a distinct and independent circulation for each body.

There is a description of a human monster, in many respects
resembling the present, given by Brugnoni in the sixth Volume of
the Memoirs of the Academy of Turin: and another by Duvernoy
in the third Volume of the Commentaries of. the Petersburgh
Academy. They had, like ours, the head, neck, and upper part
of the trunk, semi-double; four arms and four legs. They were
both formed of the union of two females. In the Turin paper no
description is given of the circulation, except that there were two

230 Proressor Ciark on a Case

hearts, enclosed in their proper pericardia, with their veins and
arteries. In the Petersburgh case, the circulation entirely differed
from ours, but yet was single for the whole mass. There were two
hearts, one much more perfect than the other: which supplied the
greatest number of arteries, as the two aortez, and received the
ereatest number of veins. The imperfect heart, after supplying
a portion of the arteries of the head and lungs, and receiving some
of the veins, anastomosed by its descending aorta, with the prin-
cipal vessel of the other heart, and by a large venous trunk with the
descending cava of the same. This case of Duvernoy, as far as
the circulation is concerned, approximates to one lately described
by Barkow, Chap. 11. No. 6059 of the Berlin Museum: two males,
with the brains nearly distinct. I know indeed of no recorded in-
stance of a human monster with a circulation nearly similar to
that which I have described. But I suspect the Turin monster
to have been such. The cireulation in a double pig, described
by Haller, Op. Min. Vol. 11. Sect. 16, very nearly resembles it.
And that by Antomarchi in a double sheep, Ann. des Sc. Nat.
Tom. xiv. is also nearly similar.

It has long been a question, whether double monsters arise
from a single germ, or from two germs accidentally united. But
the question ought to be much restricted before an answer can be
given to it. Every appearance is monstrous which is unusual: and
causes may exist capable of producing, in the uterus, an unnatural
union of twins, more or less complete in appearance and yet not
involving essential parts: capable therefore of producing a double
monster. Such slight connexions as that of the Siamese youths
may be conceived to have been thus produced, without doing
much violence to the imagination. But to suppose that pressure
so forcible should have been applied to twin germs, as to melt

of Human Monstrosity. 231

them partially into one substance, without destroying their vitality :
and at the same time so exactly applied that a single zesophagus
is neatly compacted from the divided xsophagi of two; a single
duodenum from two tortuous tubes, of which the corresponding
parts were never in the same plane: that the sterna of the germs
should have been exactly cloven in the mid line, and the divided
parts then as dextrously united, interchangably, the half of one to
the half of the other: that the aorta from one heart should, at any
period of its growth, have been violently torn away, and then
inserted into the other, the same operations being repeated on
that other: that parts so essential to life should thus be decom-
posed, and re-united under another shape, seems quite impossible.
It was in this revolting form that the difficulties of the question
presented themselves to some of the advocates of that opinion
which derives the origin of double monsters from an originally
monstrous germ. But in the present day, when the successive
steps of development are better, though not perfectly understood,
they cannot have the same weight. On the contrary the objections
are refuted by direct observation, and the suppositions on which
they are built are found to be entirely erroneous.

Before endeavouring to explain the anomalies of the present
case, it is necessary that I should refer to what is known of the
progress of the embryo from its very earliest moments. And I
undertake this task the more readily because there is no. work of
an English author which gives a detailed and at the same time
an accurate account of the process: though very valuable detached
observations may be collected from the admirable works of Harvey,
Needham, Monro, Hunter, Cruickshanks and Home. It is to
foreigners that we are indebted for accurate explanations of the
membranes of the ovum, of the modes in which the several organs

232 Proressor CiarkK on a Case

are formed, and of their early connexions with each other. And
once for all I refer below to those sufficient sources of infor-
mation *.

An ovum is an organic production of a peculiar organ, the
ovary, in the form of a vesicle, containing a matter which through
a series of changes produces a new individual capable, when it has
reached a certain stage of its existence, of supporting an mdepen-
dent life. Thus in all ova there are three parts, which are the
product of the ovary itself, viz. a nutrient matter, a membrane to
invest the whole, and an intermediate nucleus, or center, from
which the formation proceeds.

The vitellus, or yolk, the earliest part of the egg, is an im-
mediate secretion from the ovary. It is contained in the membrane
of the ovum, which is smooth within and granular externally
and is called the cortical membrane, or exochoriont. Between
these two is gradually disposed the third part of the egg. This

* Wolff. Uber die Bildung des Darmcanales, &c. mit Ammerkungen von J. F. Meckel
Halle, 1812.

Purkinje. Symbolz ad ovi avium historiam ante incubationem. Vratislav, 1825.

Pander. Diss. sistens Historiam Metamorphoseos quam ovum incubatum prioribus
quinque diebus subit. Wirceb, 1817.

Dutrochet. Cuvier. Mem. du Museum, Vol. m1. 1817.

Geoffroy St. Hilaire. Monstruosités Humaines. Par. 1822.

De Baer. De ovi Mammalium et hominum genesi. Lips. 1827.
Entwickelungsgeschichte der Thiere. Kénigsberg, 1824.
Burdach Physiologie. Leipsig, 1828.
Serres. Prevést et Dumas. Annales des Sciences Nat. Vol. x1. x11. xvi. xxi. 1827 et

subseq.
+ Dutrochet applies this term to that part of the vascular membrane of the allantois
which is outward, when that sac is doubled over the embryo and its amnion.

of Human Monstrosity. 233

appears at first as an aggregation of minute globules, which
arrange themselves on the inner surface of the exochorion. In
some one part they are accumulated in larger quantities so as to
represent a disc with a nucleus pointing towards the center of the
egg. From the interior of the yolk there proceeds a minute vesicle
containing a fluid. It moves towards the nucleus, penetrates it,
and then bursts. The fluid diffuses itself amongst the globules
of the disc, which then assumes a membranous form, and is called
the cicatricula, the germinative membrane, or blastoderma of
Pander. The central vesicle is Supposed to be the maternal por-
tion of the future germ, and when this process has taken place
the ovum is capable of impregnation.

It was long doubted where the first appearance of the ovum
in mammalia could be ascertained. The observations of Cruick-
shanks, of Burns, of Prevost and Dumas, had sufficiently proved
that it may be perceived before its arrival in the uterus: for they
discovered it in the Fallopian tubes, But, since it had escaped
detection in the ovary, the ovarian vesicle of De Graaf, in the
absence of any other ascertained rudiment, was considered to
afford the elementary fluid from which, at least in part, it is
afterwards produced. At length by the more accurate observations
of Von Baer, the ovum itself was discovered within this vesicle.
It first appears as a collection of minute granules on the inner
surface of this vesicle, and is an aggregation of the globules of the
fluid. These globules themselves are not solid, but are vesicles of
a second order. Their investing portions coalesce to form a con-
taining membrane, whilst the fluid portions become the contents
of that membrane.

The ovum presents singular varieties of size, in the different
orders of animals. In the dog it is in diameter not more than

Vol. IV. Part II. Ge

234 Proressor CLARK on a Case

ath of a line. The human ovum is still less than this: and, in
general, the size of an ovum seems to be inversely proportional
to the future evolution of the order. It may be mentioned, as
bearing upon the subject of this paper, that two ova have been
occasionally observed in one graftian vesicle, in the dog. The size
of the ovum depends upon the yolk, whose magnitude varies ac-
cording as it is intended to supply nutriment for the whole of
foetal life, or for a portion of it only, and this seems to affect the
extent of the germinative membrane. For in birds the blastoderma
occupies at first a small portion only of the surface of the yolk.
In mammalia, on the contrary, it appears from the beginning to
invest the whole. This at least is Burdach’s opinion, though not
Von Baer’s.

Constituted as above described, the ovum of mammalia escaping
from the ovary, is received into the Fallopian tube. In that organ
it does not receive any new parts. It absorbs however, as it passes,
the fluid of the tubes, and so increases in size. ‘The great changes
which it undergoes are afterwards effected in the uretus.

In the different classes of animals the successive changes in the
ova are not effected in corresponding parts of their productive ap-
paratus. In the lower forms of life the changes of the egg are
begun and completed in the ovary. In invertebral animals the
ege is so far completed in the ovary, that it receives only its ex-
ternal shell in the oviduct. In oviparous vertebral animals, as in
mammalia, the yolk and the two membranes are alone formed in
the ovary. Their ege however receives in the oviduct the suc-
cessive deposits of albumen with its membranes, the membrane
of the shell, and the shell itself: whilst in mammalia the parts
analogous to the membrane of the shell, and the shell, are com-
pleted in the uterus. Thus according as the place of the animal

of Human Monstrosity. 235

is higher in the scale, so does the addition of the accessory parts
of its ovum occur in situations more remote from the ovary.

As the human ovum passes along the tube, the germinative
membrane is more and more developed: and the activity of the
uterus is at the same time exalted. This organ secretes from its
entire surface a soft matter, which is spongy, reticulated, filamen-
tous, about the tenth of an inch in thickness. The secretion is
called by Hunter the decidua. It represents, of course, the exact
shape of the uterus, of which it is a cast, and closes all the open-
ings into that organ. Its attachment to the uterus is in the early
stages very slight, being here and there connected to its walls by
a few very delicate blood vessels. It at first merely adheres to
the uterus, as a secretion, and afterwards becomes united to it by
vessels. The ovum passing from the tube which opens obliquely
into the uterus, meets the obstruction of the decidua: this, from
its slight adhesion to the walls of the uterus, it is able to detach
for a greater or less space, and then sinks, by its weight, into
the body of that viscus. Thus as the ovum descends to the more
central region of the uterus, it carries before it a duplicature of
the decidua which supports it and keeps it in a near connexion
with the uterus. The doubled portion is called the decidua re-
flexa.. From this description it follows that at the place where
the decidua is reflected, the uterus is deprived of its covering. A
new secretion from the uterus shortly supplies the deficiency, and
this is the seat of the future placenta. The human ovum passes
into the uterus at the end of the second week after impregnation.
The reflection of the decidua begins at the end of the third.
The reflected portion receives a considerable supply of vessels
from the uterus, and soon becomes the thicker of the two. In
process of time however, from constant distension by the gradu-

GGQ

236 Proressor CiLarkK on a Case

ally increasing ovum, it becomes thinner—is brought in contact
with its external portion (the original decidua)—and finally, about
the fourth month of pregnancy, coalesces with it.

At an early period after the egg has passed into the uterus,
the granular cortical membrane expands—its granules assume an
arboresent filamentous form, and thus, penetrating the reticula-
tions of the reflected, fix it to the proper decidua and so to the
uterus. Since the ovum increases considerably in size, previous to
this attachment, the earliest nutrition of the embryo, is to be at-
tributed to absorption from the fluids of the tube and of the
uterus, by the granular exochorion. The series of changes by
which it is perfected takes place in the blastoderma. These have
not been very accurately ascertained, in the earliest stage, in the
ova of mammalia: but there is ever reason to believe that they
are exactly analogous to those which have been determined with
much care in the hen’s egg.

The first change is the gradual separation of the blastoderma
in the region of the disc into what now appear to be its compo-
nent parts, whilst at the same time it recedes a little from the
yolk. It presents an external serous layer, and an internal mu-
cous layer. At the same time the proportional distribution of
these membranes is different in different parts of the disc: thus
in the center the serous portion prevails more than the mucous,
and in the circumference the mucous portion more than the serous.
Hence the disc presents a pellucid area, surrounded by a broad
darker rmg. Between these two lamine, a third portion of the
blastoderma gradually becomes evident. It appears as a layer of
granular matter: and since this matter is the substance from
which the blood vessels and their contents are formed, it is called
by Pander the vascular layer. There are thus three parts of the

of Human: Monstrosity. — 237.

germinative membrane essentially different: and to each is appro-
priated the office of laying the groundwork of distinct portions of
the embryo, and to a certain point, of perfecting these. But no
part of the embryo is finally completed without at least one of
the other parts afterwards participating in the work.

The first perceptible indication of the embryo is a darker line
in the middle of the transparent area. It is situated on the infe-
rior surface of the serous membrane, and indicates the future
spine of the animal. On this same membrane, on either side of
the primitive streak, the dorsal plates are disposed, which meet in
such a way as gradually to form a cavity for the future spinal
cord and brain; then, on either side of these, are the ventral
plates. As the process advances, the ventral plates meet in front
only, and the head is bent downwards towards the yolk. So that
at this stage of the process, the rudiments of the whole motive
and sentient systems, are formed on the serous portion of the blas-
toderma; whilst its mucous portion, lying below the serous, is the
open abdominal space, and rests immediately on the yolk.

In the mean time the granules of the intervening vascular
layer are more developed. They are collected together more
closely in the dark border, and in smaller streaks near the pel-
lucid area. As they come in mutual contact, a change takes
place in their component parts. ‘Their external portions coalesce
to form canals which include the more fluid parts: thus, gradu-
ally, are formed on the vascular membrane a number of small
veins which anastamose and separate to form a fine net work,
and then finally open into the large external circular vein formed
similarly in the dark border, and called the ‘terminal vein. The
circle is not completed at that part which is above the anterior
extremity of the embryo, but either limb of the arc terminates

238 Proressor CLarkK on a Case

in a vessel which bends downwards and inwards to join the
heart.

The formation of the heart is absolutely similar to that of
these vessels of the areola, and takes place before the fluids in the
latter can be seen to move. By the bending of the ends of the
dorsal laminz, the blastoderma is drawn down at an angle from
its general direction upon the yolk, and forms a fold under the
head of the embryo, in which fold its serous and mucous portions
are considerably separated. This is a most important movement,
for a triangular cavity is thus left between them, which is soon
occupied by a collection of globules from the vascular intervening
membrane: and it is from this large collection of globules that
the heart is formed, by a separation of their component parts.
They become, in the same way as the vessels of the areola, a
longitudinal sac, with two posterior angles, containing a fluid
mass. The angles receive the descending vessels from the ter-
minal vein of the areola. At the very time that the heart is thus
formed, the fluid contents of the canal for the spinal cord begin
to assume a degree of solidity. The brain now consists of three
vesicles containing a fluid, which are the rudiments of the me-
dulla oblongata, of the corpora quadrigemina, and of the cerebrum.
Vhey lie, in the order mentioned, in a curved line continued from
the spinal cord: and, with other parts afterwards superadded,
may be observed before any vestige of a blood vessel can be de-
tected near them. Thus the formations of the heart, and of the
brain are synchronous, and have no connection as cause and effect.

It has been said that as the dorsal laminz are bent the ger-
minative membrane forms an angle, where it is drawn down by
the head from the general direction of that membrane. This
operation does not occur at the head alone, the same takes place

of Human Monstrosity. 239

at the whole circuit of the embryo, which now resembles an invert-
ed boat: so that the serous portion of the germinative membrane
seems to project over the dorsal surface of the embryo, in all
directions, forming a duplicature which surrounds an oval space.
The laminze of the duplicature continually grow, and advance
towards the centre of this space. This poimt they soon attain, and
then coalesce; and thus is the continuous membrane of the amnion
formed: a sereus sac which contains the embryo, and which is
incomplete only at that part where the ventral plates have not
yet coalesced, and which is the navel in mammalia, and the
point of communication between the yolk and the abdominal
cavity in other creatures.

The mucous portion of the blastoderma covers the interior, or
abdominal surface of the embryo next its spine, this disposition
being a necessary consequence of the early development of its
body on the inferior surface of the serous portion. From this
region it spreads itself over the yolk, between it, and the continu-
ally spreading vascular layer. The yolk thus covered, by mucous
and vascular membranes, forms the umbilical vesicle of mamma-
lia. It is at this moment in the place of the intestines. As the
contents of the bag become diminished by absorption on the part
of the embryo, cylindrical portions of it are folded in by the
ventral bands coalescing from before backwards, and from behind
forwards, the process first beginning towards the head. These
cylindrical portions thus form blind diverticula, which adhere
closely to the spine at the ends next the head and tail, and more
loosely in the intermediate region, where they open into the um-
bilical vesicle. As more and more of the vesicle is drawn in
and enclosed, the length of the upper and lower bowel encreases :
and the ventral plates closing more and more, the passage between

240 Proressor Ciark on a Case

the vesicle and bowel is gradually diminished, and the vesicle
projects in front of the abdomen like an hernial sac. The passage
from the vesicle into the bowel, or ductus vitello-intestinalis, is at
that part which nearly bisects the length of the future small in-
testine. In man the duct becomes gradually capillary, at length
ordinarily disappears, and then the only connection between the
vesicle and the embryo is by means of blood vessels hereafter to
be described.

Hence it appears that the groundwork of the organs of animal
life are formed exclusively on the serous portion of the germina-
tive membrane, that the system of organic life is derived originally
from the mucous portion, and that the granules of the interme-
diate vascular layer exclusively supply the matter from which the
blood vessels and their contents originate. It will be seen that these
separate portions of the blastoderma afterwards combine to form or-
gans upon which depends the advancing evolution of the embryo.

From the anterior portion of the canal of the heart, arise two
vessels, these proceed forwards, surround the anterior extremity
of the digestive canal, attain the region of the spine and then
unite to form the descending aorta. The aorta loses itself in a
small branch on the lower portion of the spine, whilst what may
now be more properly considered as its principal continuation
leaves the spine at a right angle, reaches the vascular area on
the yolk, ramifies there and inosculates with the veins of that
membrane and finally enters the terminal vein of the vascular
area. This is the first circulation: the aorta cannot be yet seen
to ramify within the body of the embryo, and a considerable part
of that body has as yet no principal trunk distributed to its sub-
stance. The artery which thus proceeds to the umbilical vesicle
is the omphalo-mesenteric artery.

of Human Monstrosity. 241

The heart by its increase soon appears to be too long for the
cavity in the neck that contains it, it becomes bent upon itself.
The mid portion of the canal forms a very acute angle with the
posterior portion which received the veins. The angle is the
future apex of the heart, whilst that part which receives the veins,
afterwards becomes the sac of the auricles. There is a constric-
tion also near its anterior extremity which denotes the bulb of
the aorta.

The two vessels which have been described as arising from
the bulb of the aorta, are not the only vessels which come from
it. And here it is necessary to record one of the most striking
anatomical dicoveries of modern times, which was made’ by
Rathké about the year 1823. He found that not only in fishes
do branchial arches and cavities exist, but that these are found
during some portion of foetal life, as in the batrachia, so also in
serpents, lizards, birds, and mammalia including man. His ob-
servations have been confirmed and illustrated by Burdach and
Von Baer. In all these animals five pairs of branchial arteries
arise from the bulb of the aorta: they do not all exist at one
time, but appear in succession from before backward. They all
arch round the beginning of the digestive canal, (the anterior ap-
proximating as nearly as possible to the rudiment of the brain,)
and coalesce in the region of the spine on either side to form a
trunk, which is a root of the aorta: for the two trunks unite to
form that vessel. These arches are separated by deep fissures in
the sides of the neck, which penetrate even into the cavity of
the imtestine. There are four pairs of fissures between the cor-
responding arches, exclusive of one in front of the most anterior
pair. The four posterior fissures on each side are gradually
obliterated, the anterior first disappearing, and the others in

Vol. IV. Part I. Hu

242 Proressor CiLark on a Case

succession. The pair of fissures anterior to the first branchial arch
afterwards becomes the mouth, whilst the bands which are bounded
by the first and second fissures afterwards becomes the lower jaw.
In man the branchial arches appear in the fifth or sixth week,
and exist only for a very short time. And here, as in the other
mammalia, in birds and in the higher amphibia, they undergo no
further development. In fishes, as is well known, they are res-
piratory organs which serve for the whole life of the individual,
and are gradually endowed with an apparatus of cartilage and
bone, admitting of numerous subdivisions on which the ultimate
branches of the vessels are spread. In batrachia, the vessels are
subdivided on membraneous productions of the soft branchial
plates, and exist only during the larva state, with the exception
of the Proteus and Siren where they are permanent.

With respect to the branchial arteries from the aorta, some
disappear as far as their stems are concerned, whilst their
ramifications persist, and others persist entirely under certain
modifications. The following is the process in the chick where,
for obvious reasons, it has been most accurately observed. From
the simple cavity of the heart, at its anterior extremity which is
the bulb of the aorta, spring in succession five pairs of branchial
arteries. They arch round the beginning of the digestive cavity,
to attain the spine, and on either side of it coalesce to form a
single stem. The stems are the two roots of the aorta, which
unite lower down in the spine to form that vessel. “The trunks
of the two first pairs soon disappear; whilst the arteries which
orignally sprung from them to supply the head are permanent,
as well as their posterior branches of communication with the
subsequent pair. When this occurs, the blood flows from the
heart, through the third, fourth and fifth pairs, to the roots of

of Human Monstrosity. 243

the aorta, and by the communicating branch on either side, fills the
arteries of the head. This third pair becomes the artery for the
neck and arm, and its communication with the bulb of the aorta
disappears also: the stream from the aorta then flowing in the
fourth and fifth pairs from the heart. The next that disappears
is the fifth branchial artery on the left side. Whilst this process
is taking place, the septum of the heart is forming, from the
apex upwards: the last part of the septum which is completed
being that immediately under the bulb of the aorta. It is a
consequence of this formation, that as the heart contracts it
sends the blood into the bulb, im two separate currents. These
currents surround each other in a spiral form as they advance
into the aorta, which arises indifferently from the two cavities of
the heart. The current from the right cavity, fills the fourth
branchial artery of the left side, and the fifth of the right side,
and these afterwards become the pulmonary arteries, with the
exception of small branches of communication with the root of
the aorta, which are the two ductus arteriosi of birds. From
the left side of the heart, the current of blood is directed to
the fourth branchial artery of the right side, which becomes the
descending aorta. This metamorphosis prevails in mammalia,
though with modifications which are not so well understood.
In the second month of the human embryo according to Bur-
dach, there are found two principal arterial stems, an ascending
stem to the head, a descending stem to the body. Of these,
the first appears to be the root of the third pair of branchial
arteries, with the subdivisions of those anterior to it, whose
roots have disappeared. Its blood comes from the left cavity of
the heart. The descending branch of the aorta comes from
the right cavity, and is composed of the remains of fourth and
HH

244 Proressor CiarRK on a Case

fifth branchial pairs. The anterior and posterior aortas are still
connected by a branch of communication, the remains of the
original root of the aorta. When the lungs are formed, the
beginning of the descending aorta supplies them with blood.
Its contents are in fact entirely distributed to them, with the
exception of what still passes through one remaining branch of
communication with the root of the aorta on the left side,
which is the ductus arteriosus: so that now the descending
aorta can only receive its blood from its branch of commu-
nication with the anterior aorta. Hence the branch of com-
munication increases, and forms between the anterior and _pos-
terior principal trunks the arch of the descending aorta. This
now derives its blood entirely from the left side of the heart,
and becomes the common origin of the ascending and descending
vessels: whilst that which was its original stem, and which arose
from the right side, now becomes the pulmonary artery.

Thus the aorta is formed from the concourse of branchial
arteries which spring from the heart, and whose direction seems
to be determined by the situation of the central nervous masses,
for they stretch towards the rudiments of the brain and then
descend in the neighbourhood of the spinal cord.

From this point the completion of the embryo requires the
co-operation of at least two of the original elements of the ger-
minative membrane.

The liver is formed by two productions of the upper part of
the intestinal tube. They proceed from the intestine as cylin-
drical prolongations of the mucous membrane surrounded by their
vascular layer: these subdivide, and form many arborescent hollow
filaments. Thus are constituted two principal lobes, which
surround the ascending vein to the heart. This vein sends off

of Human Monsirosity. 245

a loop, which ramifies with the subdivisions of the tube, and thus
the groundwork of the liver is formed, to which the parenchyme
is gradually super-added. The ramifying vein is the Vena Porte:
the original venous stem, as it advances to the heart, receives the
collection of returuing veins from the liver, and is the center to
which the ascending cava is afterwards directed.

Before the first. circulation through the vascular area is fully
established, the heart does not pulsate, but has only a vermicular
motion from one end to the other. The blood in the aorta un-
dulates, as in the dorsal vessel of insects, and has a different
direction in different parts of the tube. In this way it seems
gradually to overcome the obstructions to its direct course, and
the first circulation is established. Numerous veins of the vascular
area soon collect around the subdivisions of the omphalo-mesen-
teric artery: they form a trunk which returns to the body, in
company with that artery. They separate on reaching the intes-
tine, and the vein turns upwards, to become a principal root of
the Vena Porte. This omphalo-mesenteric vein is considered by
many as the first vessel which ramifies within the body of the
embryo.

The first circulation having been established, the arteries sub-
divide in the soft substance of the body. For instance—the
contents of the posterior .extremity of the aorta have an undu-
lating motion until the artery returns upon itself and forms a
loop: the advancing part of the loop is an artery, the returning
part a vein. From the convexity of the loop another loop is
gradually formed, and then another. So that one side of the chain
of loops at length forms a continuous arterial, and the other a
continuous venal stem; and the transverse portions of the loops
are anastomosing branches. It seems that the smaller veins are

—

246 Proressor CLARK on a Case

attracted by the larger ones, and when they come into their
neighbourhood coalesce with them. Hence the veins which
accompany the subdivisions of the anterior aorta, and the in-
tercostal arteries are guided by these vessels directly to the
venous sac of the heart, whilst the same sac of the heart is
the centre of the veins of the abdomen and lower parts of the
body. The difference is that these two series of veins attain the
heart most directly, the one by following the course of the ar-
teries, the other by diverging in some degree from that course.

But to return to the membranes of the ovum, which hitherto
we have described as an external exochorion, and an internal
serous amnion.

The allantois is a production from the posterior part of the
bowel at its lower extremity. It proceeds from the cloaca, (for
as yet there is no urinary bladder,) as a thin transparent whitish
vesicle, through the unclosed abdomen at the part nearest to
the tail, whilst the umbilical vesicle passes out nearer to the
head. It consists of two layers of the blastoderma, the mucous
and vascular layers. Its existence is later than that of the liver,
but earlier than the production of the arteries which form the
umbilical cord. It is observed in the human embryo in the
third or fourth week, but here it does not attain any great size,
and disappears in the sixth: whilst in many other mammalia,
as also in birds, it is reflected over the whole amnion, which with
the embryo it includes as a sac doubled on itself, the sac being
somewhat distended with fluid. When the urinary organs are
developed, the bladder is formed out of that portion of the allan-
tois within the abdomen which is connected with the intestine.
This at first does not increase in size with the rest of the sac,
but appears rather as a duct connecting the bowel and the ex-

of Human Monstrosity. 247

panded outer part of the allantois. As it expands, the terminating
branches of the aorta, follow the vascular external layer, and run
upon its sides, and so pass out of the abdomen. In birds, and
the higher amphibia, the umbilical arteries ramify most minutely
on the external vascular layer, particularly on that portion of the
doubled bladder which is next the exochorion. In man _ the
allantois external to the abdomen soon dwindles to a capillary
tube, its original vascular layer still continuing as a conductor
of the umbilical arteries. In the other mammalia the allantois
persists throughout foetal life, and receives numerous branches
from the umbilical arteries. The veins which return the blood
are collected into one stem which accompanies the arteries as
far as the navel. There the two orders of vessels separate. The
veins pass as one large trunk to the general venous center, the
ascending cava, below the heart. This is the principal stem:
subordinate branches are given to the liver as it passes between
its great lobes, and a branch of communication to the Vena Porte.

In man the umbilical arteries are immediately surrounded by
the vascular membrane which originally belonged to the allantois.
When the cord reaches the placenta, now beginning to be formed,
that membrane spreads itself over the whole inner surface of the
exochorion, and forms its vascular portion. Thus is the chorion
formed ; it consists of an external villous exochorion, and an inter-
nal vascular endochorion. In the case of twins the two embryos
have usually a common exochorion, with the other membranes
proper to each. This is an instance of the law which has been
well elucidated by M. Serres, viz. that in the primitive state of
embryos when two homogeneous organs are brought in contact,
they coalesce and form one. If the amnion be also common to the
two, the case, though not uncommon, is pathological, or monstrous.

248 Proressor CriarK on a Case

The human placenta is developed when the umbilical vesicle
begins to disappear: and what it is that determines the part of
the uterus on which it is formed has been already described.
The subdivisions of the umbilical vessels distribute themselves
essentially to those tufts of the exochorion which are in the region
where the decidua was reflected. The tufts gradually grow to-
gether in the third and fourth months of pregnancy, and thus
the foetal placenta is smooth on its inner surface and presents
numerous inequalities on the external: which inequalities are the
several collections of the prolonged and subdivided villi of the
exochorion. On every one of the villi as a stem, and on all
their subdivisions as branches, are distributed corresponding sub-
divisions of the umbilical vessels which directly anastomose.

As the abdomen gradually closes the only opening left is the
navel, which is near the tail end. The umbilical cord consists
externally of a layer from the amnion, which passes from the
cuticle of the foetus: beneath this is the vascular sheath originally
derived from the allantois, which on the one hand is lost in the
aponeurosis of the abdominal muscles, on the other in the endo-
chorion. It immediately surrounds the vessels, as well as the layer
of cellular membrane which they derive from the peritoneum, and
which with the vessels contains also a gelatinous fluid, gradually
to be absorbed by the foetus, and called the gelatine of Wharton.
Thus the original external membrane of the ovum, the exochorion,
has at no time any direct connection with the cord, or with the
embryo*.

* Besides these parts the cord at the earliest period of its formation, when it is wide
towards the abdomen, contains a loop of intestine, the umbilical vesicle, and the allantois.

of Human Monstrosity. 249

The lungs in all vertebral animals are in their origin probably
processes from the mucous layer of the abdominal canal, with
corresponding accompanying portions of the vascular layer, and
are in their earliest stage analagous to the swimbladder of fishes,
and the lungs of the Proteus and Siren. They are begun in the
sixth week, and receive their further development in proportion as
the other respiratory apparatus disappear. These latter are the
branchial arches, and endochorion of the allantois with the pla-
centa. These three last respiratory organs do not appear to be
intended to perform their functions perfectly at the same time m
any class of animals. When the branchial arches of birds and of
the mammalia disappear, then the allantois is the active respira-
tory organ: and where the arches are permanent, as in fishes for
the whole of life, and in frogs until the lungs are developed, the
allantois is not formed at all.

Ir is now time that I should declare my opinion respecting
the mode in which the appearances, which the subject of this
paper presents, have been produced. And this I do with much
diffidence, though it has not been hastily, nor carelessly formed. I
look upon it, then, as a case of twins, whose germs were originally
perfect and distinct, notwithstanding all the difficulties which that
opinion includes. The two ova I suppose to have been brought
into close contact in the Fallopian tube, and that thus their exo-
chorions have coalesced, as is usual, to form a common cortical
membrane: and that, besides this, a coalition has also been ettected
between corresponding points of the germinative membrane, near
the disc. The nearly perfect symmetry of the two faces, and of

Vol. VV, Part I, | Lori

250 Proressor CLark on a Case

the whole united portions of the two bedies, shews that if such
an union did take place it must have been in corresponding parts.
These parts I conclude to have been in front of the primitive
streaks of the embryos. Let us now suppose that the evolution of
each embryo advances, that the dorsal plates and spinal cavities
and vesicles of the brain are formed in each: until the space
between the anterior extremity of each is occupied by the anterior
vesicle of the brain of each. These, according to M. Serres’ law,
will now coalesce, and our dissection shews that these are the
only parts which did coalesce, for the cerebella and corpora qua-
drigemina were separate and distinct. The ventral plates of the
vpposite embryos necessarily meet in front: and there is an effort
to bend the head downwards to form the cavity for the heart.
This bending takes place in the direction of the ventral plates,
the effort being resisted in every other. Thus there are cavities
for the hearts properly formed: and they receive their blood from
opposite sides of the united vascular areas. Let us now consider
how the aortas are to be formed. Five pairs of branchial arches
project from each heart on either side of the neck. They arch
round to the basis of the brain to reach the situation of the spinal
cord, and there coalesce to form the roots of the descending
aorta. Thus the descending aorta will, in this case, be formed by
branchial arteries which come from different hearts: they are the
root of the third pairs, which arches from either heart in a di-
rection from right to left, until it gains the spine. Thus all the
arteries which arise below the original third branchial artery,
come from the descending aorta, whilst the arteries of the head,
come from the arch which connects the descending aorta to the
heart. And that distribution is exactly the one before us: the
aorta has reached the spine to which it does not seem to belong,

of Human Monstrosity. 251

because it is formed according to the usual laws. The veins also
follow the usual Jaw ;—the superior caya is guided by the arteries
(vid. page 28) of the arch of the aorta to the venous side of the
heart; and the inferior veins—(those which return the blood of
the descending aorta of the other heart)—to the same cavity. It
must be recollected that this venous center is determined before
there are any limbs and before the aorta has subdivided in the
body, and that it is quite independent of the latter vessel. There
is one apparent difficulty: each foetus receives its right subclavian
artery from one heart, and returns its right subclavian vein to
the other: whereas the left subclavian artery and vein are both
attached to the same heart. But this also comes under a general
rule when it is remembered that the subclavian veins join the
jugulars to form a descending cava, and that the course of the
jugulars is determined by the carotids, whilst the subclavian
arteries are originally from the extreme upper part of the de-
scending aorta.

The sac, by whose intervention the bowels of the two bodies
are united, appears to me to be the most interesting feature of
this case. I consider it to have been developed from the umbi-
lical vesicles of the two embryos, which, according to my hypo-
thesis, were united at their anterior parts: I believe it, in fact, to
be the conjoined vesicles. It will be remembered that the vesicle,
in the normal condition, at first fills the abdomen: that as the
bowel is formed at its expense, it gradually retreats into the cord,
being borne onward by the continually increasing loop of in-
testine, and that at length the duct of communication between
the bowel and vesicle vanishes, and the vesicle itself rests at the
farther extremity of the cord, or between the chorion and amnion

in front of the placenta. In the united bodies the circumstances
112

252 Proressor Ciark on a Case

are entirely different: the vesicle has four points of attachment
instead of two, it is pressed by equal forces in opposite directions,
and is kept at rest. Hence the upper coils of small intestine
were found much compressed with a very short mesentery, and
closely connected to the walls of the vesicle: whilst the lower
bowels were disproportionately developed, and the mesentery
much extended between them and the spinal columns. If this
explanation be accepted it sets the question at rest concerning the
existence of an open duct of communication between the umbilical
vesicle and the small intestines in the human embryo, a commu-
nication which has been denied by great authorities, by Emmert,
Heechstetter, and Cuvier. In a rare case, (Tiedemann, Anatomie
der kopflésen Missgeburten, tab. rv.), there was found a true um-
bilical vesicle attached to the intestinal canal. The foetus had
arrived at the full term, and was in many respects monstrous.
In Brugnoni’s case it is very remarkable that the single duodenum
terminated in a vesicle, whilst the lower portion of the small in-
testines of each foetus were not connected with it, but commenced
as blind tubes. -

The upper part of the intestine, that from which the cso-
phagus, stomach, and anterior portions of the small intestine are
afterwards formed, was perhaps probably at its origin, and from
the first, a single tube connected to the spine of each embryo.
That it was formed from the parts appropriated to each I con-
clude to be a fact: because I observe that there are two pairs of
lungs with a trachea for each, and two livers. The appearance
also of the stomach and the presence of two spleens seems to in-
dicate that it has been formed from the rudiments of what ought
to have been two. The existence and the magnitude of the liver
on the left side of the drawing Pl. 14. shews that the unintelli-

of Human Monstrosity. 253

gible mass between it and the spleen was once intended for duo-
denum, though never matured into that viscus.

IT admit that there are difficulties in the explanation which I
have thus given; but I advance it because it is only by a careful
consideration of individual cases that the mystery of these unusual
productions can be unravelled: and what is yet obscure in organic
development be satisfactorily explained. Such productions as that
which I have described were formerly kept in museums as objects
of idle wonder. They are now closely investigated as particular
cases of a general problem: cases in which some of the conditions
are modified or suppressed, and of which the value and force are
therefore rendered apparent. And thus do they present ready to
our hand, means for the solution of what is most difficult—devices
which no human ingenuity could have contrived.

254 Proressor Ciark on a Case

DESCRIPTION OF THE PLATES,

which are accurate orthographic projections of the original, drawn with
Mr. Wiuu1s’ Machine.

PuateE 11

Contains an anterior and a posterior view of the subject, half the real size.

PLATE 12.

A. Front view of the cranium and face, where is a rudiment of the zthmoid bone over
the orbit. Here there are two frontal bones. Between the upper and lower
jaws the tongue is seen. The bones, which make up the orbit in this face, are
lettered in the plan G.

B. Cranium and face on the opposite front, where the frontal bone is single. The
bones which make up the orbit are lettered in Fig. H.

C. View of the vertex.
View of the occiput, face 4 to the right hand, face B to the left hand.

The proboscis with its skin projecting above the orbit, where the soft parts have
shrunk. ;

F. ‘The bone of the proboscis where the skin of the same is turned down upon the

orbit.
G. a. Frontal.
6. Sphenoid.
c. Maxillary.
d. Malar.

H. Letters correspond to those of G: the orbitar frontal plate single.

boyny fropog

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maMewruaprr 2D Y bon Srryn9 ;
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2." : r * :
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MAKLOGULUAOS OMIA O06

MONA

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of Human Monstrosity. 255

m. View from above of a pair of united Ingrassial processes of opposite sphenoid
bones, under (m) passes the optic nerve. There is an open fissure in the

mid line.
nm. View of the same from below.
0. Lateral view of the same.

PLatTE 13.

I. View of the intestines.
K. View of the common basis of the skull.

L. Map of the same. The portions of the occipital bones are all yellow: of the
temporal bones red: of the frontal bones brown: of the sphenoid bone, with
four great ala, and umited ingrassial processes between two of those ali on
each side, blue.

PuLaTE 14.
The Circulating System.

M. Represents the bodies drawn asunder as far as possible, after the diaphragm has
been removed and the hearts separated. The bodies are in the plane of the
paper, the head and neck below that plane. a a, the inferior cava broken
off near the point of its entrance into the heart, as is the azygos of this body—
the Aorta of this body is from the other heart, &c. vid. page 9. The Cso-
phagus, and the trachea of the lungs attached to the lower heart, are seen
passing in the space between the two hearts.

N. Shews the Arteries from the arch of the Aorta on one side—the veins, which form
the descending cava—portions of the ascending cave, and of the descending
aorte. The two hearts are seen, and portions of the lung between them. This
is the same view as would be presented if the upper heart in Fig. M, were
turned down, to its proper position, and the head and neck raised

VIII. On the Examination of a Hybrid Digitals.

By THe Rev. J. S. HENSLOW, M.A.

PROFESSOR OF BOTANY, AND SECRETARY TO THE CAMBRIDGE
PHILOSOPHICAL SOCIETY.

[Read Noy. 14, 1831.]

Atruoucu the propagation of hybrid plants has been much
attended to of late years by several Horticulturists in England,
their experiments, for the most part, seem to have been under-
taken for the sole object of encreasing the forms of beautiful
flowers, or of modifying the flavour of delicious fruits. But the
more curious and important physiological facts elicited by the
phenomenon of hybrid productions do not appear to have received
a proportionate degree of attention from those who have been
engaged in these experiments. Chance having favoured me with
a hybrid Digitalis during the past summer (1831), in my own
garden, I employed myself, whilst it continued to flower, which
was from June 19 to July 22, in daily examining its characters
and anatomizing its parts of fructification. I was careful to com-
pare my observations, with as much patience and accuracy as
I can command, with the structure of its two parents. It seemed
to me not unlikely that something interesting might result from
a rigorous examination of this kind, or at least that its recorded
details might serve as a point of departure for future observa-

tions.
Vol. IV. Part II, K x

258 Proressor HENstow’s Examination

The plant in question was undoubtedly a seedling from a
specimen of D. lutea. I have this species and D. purpurea
alone of the genus cultivated in my garden, where several
plants of each had been allowed to scatter their seed, and the
seedlings to grow wherever they chanced to come up. I had
already remarked a singularity in the general appearance of one
of these, and was watching the expansion of its flowers, when
I was agreeably surprized to find it to be a decided hybrid,
obviously having most of its characters exactly intermediate be-
tween those of purpurea and lutea. I had no doubt whatever
of its beimg a seedling of lutea, from the position which it
occupied in the garden: in coming up amidst several plants of
this species in a spot where an old plant had grown the year
before; neither had any plant of purpurea grown in the same
border. Besides which, my plant exactly agrees in most par-
ticulars with a hybrid procured by Koelreuter in 1768 from seeds
of lutea fertilized by the pollen of purpurea*. His account is ac-
companied by a rude and inaccurate figure which by no means
tallies with his own description of the plant. In general habit, this
hybrid approaches much nearer lutea than purpurea, Plate xv.
Fig. 1. It is however decidedly taller and more robust than any
specimens of the former species which my garden ever produced.
Koelreuter indeed asserts that the specimens raised by him were
taller than either of their parents, but he assigns a: lower limit
to the height of purpurea than that to which many plants of
this species have attained with me. Notwithstanding its more
robust character and somewhat darker hue, the eye would
scarcely have recognized, upon a mere casual observation and

* Acta Acad. Petropol. Anno 1777.

of a Hybrid Digitalis. 259

before its flowering, any peculiarity sufficiently striking to class
it apart from some of the varieties of lutea, but a little closer
inspection immediately detected certain decided points of dif-
ference. The whole plant is not so smooth as lutea, having a
decided tendency to become downy, and being completely so
on the under surface of the leaves, Plate xv. Fig. 2. The gla-
brous surface of lutea is one great characteristic of the species;
though, if the D. rigida of Lindley* is to be considered as a
variety of it, which he seems to think probable, even this character
fails. A few hairs are always indeed distributed here and there
in the ordinary state of this plant, and seem to indicate the
possibility of a transition from the one condition to the other,
dependant probably on certain circumstances of soil or situation.
From the ordinary condition of the leaves of lutea, however,
those of the hybrid differ in a marked manner. They are even
nearly as woolly on the under surface as the leaves of purpurea.

Examination of the external characters of the Hybrid.

I shall first describe the external characters of its several
organs, comparing them with those of the parent plants. In
Plate xvi, the corresponding parts in the fructification of the
parents and of their hybrid are arranged in three columns, those
of the latter occupying the middle column. A single glance of
the eye will thus be sufficient to shew how exactly intermediate
most of its organs are both in size and form, and in some cases
also in color, to those of the two parents. There are however
some remarkable deviations from this condition, which will be
presently noticed.

Ph. SS eee a ee

* Lindley Digitalium Monographia, fol. Lond. 1821.
KK2

260

Proressor HENsLow’s Examination

Comparative view of the external characters of the three plants

represented in Pirates XV. and XVI.

Purpurea.
Biennial.

3—5 feet.
13— feet.

less secund, and
laxer.

woolly.
very soft.

crenato-dentate.

petiolate, oblong.

broader and shorter.

longer than the
Calyx and fre-
quently than
the bracteas.

large, cernuous.

more spreading.

broader.

Hybrida (purpureo-lutea*).

PLATE XV.

Root. Perennial, according to Koelreuter,
and apparently so in the present instance,

the plant having thrown out several offsets.
Stem. About 3} feet.
Raceme. About 14 feet.

secund, dense, nodding above.

Leaves.
below.

Dentate.

radical, sub - petiolate, broadly -lanceolate,
Fig. 2.

caulinar, sessile, narrower.

Nearly smooth above, quite woolly
Somewhat soft.

Bracteas ; Lanceolate.

Pedicels. About the length of the Calyx,
and and somewhat shorter than the brac-
teas.

Flowers, medium size, nearly horizontal.

PuatE XVI.
II.

1. Calyw, moderately spreading in flower,
afterwards connivent.

a. sepals, ovato-lanceolate, the odd one much
narrower.

Lutea.

Bi-tri-ennial.

2—3 feet.
$—1il feet.

denser.

glabrous.
firmer.
dentate.

somewhat nar-
rower.

narrower & longer.

shorter than the
Calyx and much
shorter than the
bracteas.

small, more
drooping.

III.

less spreading, at
length more
closed.

narrower.

* If a general rule for naming Hybrids should be thought advisable, perhaps it will
be found convenient always to prefix the name of the plant which supplies the pollen to
that which furnishes the ovule.

Purpurea.

more hairy.
purple.
spots more nume-
rous, deep purple,
and rings paler.
less hairy.
obscurely 4 lobed,
the upper emar-
ginate.
half the length,

convergent.

deeper orange-yel-

low, with nume-
rous spots often
confluent.

muchmore oblique.

few hairs.

much more acute.

more ovate and
more. pubescent.

much more nume-
rous.

of a Hybrid Digitalis.
Hybrida ( purpureo-lutea.)

b. hairy on the margins.
2. Corolla, Yellow ground tinted with red.

A few dark purplish-red spots surrounded
by a paler ring in the throat and tube.

Smooth, with hairs in the mouth.

Distinctly 4 lobed, the lobes blunt,
uppermost notched.

the

3. a. Stamens length of the tube, nearly
parallel.

b, ec: Anthers yellow inclining to orange, with
a few small scattered purple spots.

Oblique to the filament, converging above.

d, e: Pollen White, elliptic when dry, and
sperical when moist. Some of the grains
obscurely three-cornered, many are abortive,
but those perfected are of exactly the same
size and shape as in purpurea and lutea,
being somewhat less than +, of an inch in

diameter.

4. a. Pistil, covered below with small glandu-
lar hairs.

style cylindrical, with a few hairs on the
lower part.

b. stigma cloyen, very obtuse.
ce. ovarium oblong, pubescent.

d. ovules numerous, and exactly of the same
shape and size as these of purpurea
and lutea.

261

Lutea.

less hairy.
yellow.

no spots.

more hairy.

4 lobes deeper,
acute, the upper
deeply notched.

somewhat more ex-
tended and diver-
gent, according
to Koelreuter ;
but I could see
no very appreci-
able difference.

lighter yellow, no
spots.

hairreacheshigher
up.

more acute.

more acute and
less pubescent.

much less nume-
rous.

262 Proressor Henstow’s Examination

Commentary on some parts of the preceding comparison.

Raceme. Although one of the characters of lutea lies in the
very decidedly secund position of the flowers, some plants have
them disposed in a squarrose manner round the axis.

1. Calyx. About one half the number of the flowers of the
hybrid had five sepals and the other half six, (Plate xvr. II. 1. c.)
and the sections given (from d. te 7.) represent the different modes
of their arrangement. Figs. d. and h. however appear to be
their normal condition in estivation, the other modifications having
probably resulted from inequalities introduced during the expan-
sion of the flower. The occasional development of a sixth sepal
seems to be no uncommon occurrence in this genus, and I have
met with it several times in specimens of lutea and ferruginea.

2. Corolla. In the colored copies of Professor Lindley’s mono-
graph, there are two varieties of lutea (see his Plates xxiv and xxv)
in which the corolla is tinged with red. One of these (Plate xxtv)
he considers to be a hybrid plant. In shape and size it ap-
proaches very nearly to the subject of the present paper, but
the other (Plate xxv) more closely resembles lutea. In his figure
of lutea also, (Plate xxii1) there is a little tinge of red in the
mouth of the tube, on each side the base of the lip. I have
never myself found the slightest tinge of red in any specimen
of lutea, though the yellow is deeper and more inclining to
orange in the parts above mentioned. If however it should be
quite certain that genuine specimens of lutea do occur with a
tinge of red in any part of their corolla, this circumstance must
considerably modify our speculations as to how far the present
hybrid may have derived this color from the male parent.

of a Hybrid Digitalis. 263

Flowers of lutea are not unfrequent with the lower lip notched
(Fig. y), which indicates the presence of a supernumerary petal
blended into the tube of the corolla. In about half a dozen
instances I even found this petal quite free, (Fig. 8) and I be-
lieve occupying the same position as the sixth sepal in the
anomalous cases just referred to. In D. ferruginea, however, I
have sometimes found a sixth sepal and a notched lip in the
same flower. These anomalies may therefore be considered ana-
logous phenomena among the supernumerary developments of
the two organs.

3. Pollen. In comparing the action of the three pollens when
immersed in water, I observed all the phenomena usually at-
tendant on this experiment, to take place in those of purpurea and
lutea: their grains quickly swelled and their granules were ex-
ploded in the form of a dense cloud (Fig. F and %). Two kinds
of granules were also observed, the smallest and most numerous
of which were too minute for me to be able to ascertain their
precise shape and dimensions by the highest powers of my in-
struments; the others, much fewer in number, were considerably
larger, and lay dispersed among the smaller like pellucid spots
on a darker ground; and these might even be distinguished
through the coats of the grains before their expulsion had taken
place. Some pollen of purpurea taken from a withering stigma
exhibited very distinctly the presence of the exserted mem-
branous tubes (boyaux) described by A. Brogniart, Amici, and
others, in the Ann. des Sciences, (Fig. G). Some of the granules
also were marked on the surface by three blotches (Fig. H).
Grains of pollen taken from the hybrid readily swelled upon
immersion in water, though most of them appeared to be void
of granules. Some few however certainly contained the larger

264 Proressor Henstow’s Examination

kind of granules, and I could see their explosion accompanied
by successive and sudden contractions and dilatations of the
grains themselves. But I could never detect any cloud of smaller
granules similar to that which was exploded from the pollen of
the parents, and which always proceeds from the grain by a
continuous and slow emission, whereas the larger granules in
the hybrid were discharged at intervals, and by separate efforts,
and lay scattered at a distance from each other over the field of
view (Fig. f.)

Koelreuter has given it as his decided opinion, derived from
his numerous experiments, that true hybrids never reproduce their
kind. Later experimenters have doubted this fact, and some seem
to consider the question as quite settled to the contrary, at least
with respect to the possibility of fertilizing a hybrid by the pollen
of one or other of the parent species. But in prosecuting this en-
quiry we must be very cautious to keep in view the _ perfect
distinctness of the two questions, whether it be probable and
whether it be possible that hybrids should reproduce their kind.
If it be possible that a true hybrid may do so, it may still be
very improbable, from some deficiency in that connection of cir-
cumstances, of whatever description it be, which is essential
to secure the fertilization of the ovule. We might imagine* for
instance, so great a discrepancy to exist between the respective
circumstances suited to the healthy action of its vegetative and
reproductive functions, that although one climate may be adopted
for securing the former, another might be required for obtaining

* This hypothesis is thrown out merely in the way of illustration, and not as likely to,
afford any solution of the cause of infertility observable in Hybrids, at least in most of
them.

of a Hybrid Digitalis. 265

the latter, and thus the plant might continue to grow and flourish
in one latitude, and yet be incapacitated for ripening its pollen
or perfecting its ovules unless it could also thrive upon removal to
another. There are certain plants, considered to be hybrids, which
undoubtedly reproduce their kind freely enough; but some of
these at least, if not all of them, are mere varieties of the same
species. Thus Koelreuter ascertained that all the plants raised
between D. purpurea and D. thapsi, by fertilizing the ovules of
either by the pollen of the other, were constantly prolific, but
then he also ascertained that D. thapsi itself when cultivated by
him, after five generations assumed all] the characters of purpurea.
IIe consequently rightly inferred that D. thapsi was to be con-
sidered no otherwise than as a Spanish variety of the more
common form of the species. If, again, it were possible for a
true hybrid to be fertilized by the pollen of either of its parents,
though it could produce no fertile pollen for itself, it would
then evidently be in much the same condition as the female
plant of any dicecious species, and its fertility might be secured
by the instrumentality of insects, &c. In the present plant I
repeatedly observed that the blossom always fell before the an-
thers on the shorter stamens had burst; and in order that this
should not operate in diminishing the chance of impregnation, I
touched some of the stigmas with the pollen extracted from
these anthers, but without any success. Possibly however the
pollen was not sufficiently ripened. I also touched other stigmas
with the pollen of purpurea, and others again with that of lutea;
but all these experiments failed in fertilizing any of the ovules.
Koelreuter was equally unsuccessful in his attempts to fertilize
this hybrid. I must here record what has appeared to me
a remarkable circumstance, brought before my notice during
Vol. 1V. Part II. Li

266 Proressor HEenstow’s ELaamination

the prosecution of these enquiries. There were three or four
plants of lutea in my garden which were quite deficient in
pollen, and which nevertheless produced perfect seeds. I was
unable to detect even a single grain of pollen either healthy
or abortive in their anthers, though these latter organs appeared
to be well formed and perfected. The ovaria of these plants
indeed contained plenty of ovules, most of which I afterwards
observed had been fertilized, since their seeds ripened. These
plants must therefore have been fertilized by the pollen of other
specimens in their neighbourhood ; at least according to all our
present notions on this subject. But then the ovules of the
hybrid were also similarly circumstanced, and if they had been
capable of receiving the same influence from other plants, there is
no apparent reason why they should not haye proved fertile, also.

4. Ovules. In the parent plants, the ovules begin to grow
and develop themselves immediately after the fall of the corolla,
whilst in the hybrid they soon wither away. It is remarkable
however, that all symptoms of decay in the ovarium are strictly
limited to the ovules themselves, for even the little protuberances
upon which they are seated on the placenta remain succulent,
as do the various parts of the pericarp, including also the base
of the style: all which continue healthy and attain their perfect
dimensions, the valves alone slightly collapsing from the deficiency
of the ovules in the enlarged cells. Plate xvi. Fig. 4. But the
stigmatic tissue dries up, and a cavity is thus left through the
upper part of the dissepiment, forming an opening between the
two cells, Fig. 5. e. The same effect sooner or later takes place
also in the seed vessels of the parents.

Recapitulation. Tn reflecting upon the points of resemblance and
of disagreement in the organs of fructification of these three plants,

of a Hybrid Digitalis. 267

the most striking circumstance which we have hitherto noticed in
their external characters, is the perfect identity in size and shape
both of their pollen and of their ovules. As the respective organs
which contain these bodies, viz. the anthers and the ovaria, are
each proportionate to the different sizes of the three flowers them-
selves, it is evident that a flower of lutea must have much less
pollen and many fewer ovules than° one of purpurea, which in
fact the most casual observation is sufficient to shew. The ovules
of the hybrid also are about intermediate in number to those
produced by the parents. It will be a subject worthy of future
investigation, to determine whether one condition necessary for
securing the hybridity of two species, require their pollen and
ovules to be of the same, or of nearly the same dimensions. Ex-
cept in the above instances, and in the very peculiar shape of
the stigma, all the other external characters of the hybrid ap-
pear to be precisely intermediate between those of its parents.
The chief physiological difference observable in the external
economy of the organs of fructification seems to reside in the
fall of the corolla, which in the parents does not take place
till after the anthers have discharged their pollen and become
perfectly withered, whereas in the hybrid the corolla falls before
the anthers on the shorter stamens have burst, and when even
those on the longer pair, although opened, have hardly parted
with their pollen, and have not as yet become in the least withered.
The style and stigma of all three appeared to comport themselves
alike, that is to say, they all began to wither soon after the fall
of the corolla. |

LL2Q

268 Proressor Henstow’s Examination

Examination of the internal structure of the Organs
of Fructification.

Before I begin the detail of this examination, I may at once
state, that so far as I have hitherto been enabled to pursue it,
I have not perceived the slightest difference between the internal
structures of the three plants; and as their organization is some-
what different from any of the cases selected by Mons. A. Brogniart
to illustrate his paper on the formation and developement of the
embryo, the’ present attempt may not be without some general
interest to the physiologist, independent of the objects connected
with the particular enquiry for which it has been undertaken.
The method which I pursued was always to examine the various
parts dissected, first, in specimens of purpurea, and then to com-
compare them with the like parts in hybrida, and lutea. Though
it is possible therefore that I may accidentally have overlooked
some defect and dissimilarity in the internal structure of the
hybrid during this common and simultaneous examination of all
the three, and may have represented in the drawings some ap-
pearance or other strictly belonging only to the anatomy of pur-
purea, yet I do not think such an error could very probably
have occurred. As the main object in view was the direct
comparison of the three plants, any striking difference at least
would have been noticed, and the subject have been submitted
to a rigorous re-examination.

Vessels of the Pistil. Plate xvii. Fig. 1. represents a longi-
tudinal section of the ovarium perpendicular to the dissepiment,
and consequently passing through both the cells; and Fig. 2. is
another longitudinal section, at right angles to the last, and through
the plane of the dissepiment, or rather, it represents the surface

of a Hybrid Digitalis. 269

obtained by tearing the ovarium asunder down the thickness of
the dissepiment, which is composed of two skins with parenchy-
matous matter between them. The threads of vascular tissue
arranged in a circle round the axis of the pedicel (a), after giving
off veins to the calyx and corolla (4), and again to the pericarp (c),
diverge on either side into the placenta (d), a little above its
lowest point, and then ramify or subdivide through its substance
into separate fibres (d’) which proceed directly to the bases: of
the ovules. Fig. 3. represents a transverse section of the upper
part of the ovarium with the lower part of the style; the valve
which is nearest the spectator being removed, as also are the
ovules in this cell. The smaller veins (c'), of which more than
twenty are seen rising through the pericarp, all terminate in the
base of the style; but the two larger ones (c), which run along
the loculicidal edge of the pericarp, rise through the whole
length of the style. The stigmatic tissue (e), (Fig. 1. 2. 3.) de-
scends down the middle of the style till it comes into contact
with the summit of the placenta. When the appearances here
represented are examined with the highest magnifiers, their more
intimate structure is exposed, as in Plate xviit. where Fig. 1.
and 2. are two transverse sections of the pistil, of which the
former corresponds to one quarter of the circumference of the
ovarium represented in the lower part of Fig. 3. Plate xvil.,
and the latter agrees with the section through the style in the
upper part of the same figure. Plate xvi. Fig. 3. and 4. are
longitudinal sections of the same organ, the former through the
stigma, the latter through the summit of the ovarium where the
stigmatic tissue (e) descends to the placenta, as in Fig. 1. Plate
xvit. In these highly magnified sections all the corresponding
parts are designated by the same letters as in the former figures.

270 Proressor HeENstow’s Examination

The veins (c), (d), &c. are in all cases composed of bundles of
trachew, which in the larger veins (c) are very numerous. I have
counted sometimes between thirty and fifty combined in the con-
struction of a single vein (c), a fact which would not be suspected
upon a casual observation, but which becomes evident by digesting
the style in nitric acid, when these elementary parts are easily
separated. Their termimations are in the form of elongated cones,
and they all end together, a short distance below the stigma. (See
Plate xvi. Fig. 3.) The other elementary parts of all these veins
are certain extremely delicate tubes which invest the central bundle
of trachez, and give it the appearance of being surrounded by
a mucous or glutinous substance, but which under the highest
powers of the microscope may be separated into these tubular
vessels, whether subdivided or not by transverse diaphragms, I
was unable to satisfy myself. This very delicate tissue has the
same general appearance as the stigmatic tissue, which in these
plants descends down the centre of the style, to the summit of
the placenta. Where this latter tissue terminates in the stigma, it is
indeed evidently composed of distinct cells, easily separable from
each other by nitric acid, Plate xvii. Fig. 3. (g). Lower down
however the cells are more elongated (7), and lower still, where
this tissue meets the placenta, I could neither detect any
transverse diaphragms in it, nor even detach its cells (if they
were such) from each other at their extremities by ‘the action
of nitric acid, though they were easily separated longitudinally
into long filamentous strings. In this part of its course therefore
the stigmatic tissue appears rather to be tubular than cellular in
its structure. After this tissue. has become divided into two bands,
penetrating on ‘either. side through the dissepiment into the two
cells, it seemed to me, upon a most careful examination, to coat

of a Hybrid Digitalis. 271

over the whole surface of the placenta. It is very difficult how-
ever to be quite certain of this fact, and I may be wrong; but
after numerous dissections made upon the three plants, I found
I could generally raise, with the point of a very fine needle, a
thin gelatinous film of a delicate fibrous structure from between
the ovules Fig. 4. (e’), which film seemed to be similarly con-
stituted, and also continuous with the stigmatic tissue (e).

Cellular tissue of the Pistil. These cells are for the most part
compressed into tolerably regular rhomboidal dodecahedrons, ex-
cepting in the placenta, where, as the ovarium increases, the vesicles
assume that irregular character so well described and represented
by Mons. A. Brogniart in the parenchyma of the leaf, (Ann. des
Se. Vol. xx1.) and they have the same sort of interstices filled with
air between them as those which occur in that organ. When
the style is digested in nitric acid, the separate vesicles of its
cellular tissue become cylindric-oval, Fig. 5. (0): and I have repre-
sented an appearance (p) which was noticed several times upon
some of these vesicles, of a faintly marked band running down
one side.— Further examination may perlaps throw some. ad-
ditional light upon this circumstance, but at present I know not
to what cause it may be ascribed.

Epidermis of the Floral Organs. Plate xvii. Fig. 6, 7. The
flattened cells are of the same size in the three plants, their
diameter being somewhat more than the thousandth of an inch.
They vary in shape from hexagonal to quadrangular prisins
bordered by straight, or waved sides. This membrane is irregu-
larly supplied with stomata (7). When digested in nitric acid,
the cells assume an appearance represented in Fig. 7., as though
the granular matter they contain were coagulated into a nucleus, or
else were enclosed in a separate internal vesicle. Whether this

272 Proressor HeENstow’s Examination

appearance originate in any optical deception, I could not
sufficiently satisfy myself; but if, as I am inclined to think, it
does not, the fact must have been hitherto overlooked from the
difficulty of detecting the true plane of junction between the
contiguous cells, owing to the very great transparency of their
membrane. Thus, in Fig. 6, where this epidermis is less mag-
nified, the cells appear to be separated from each other by
anastomosing veins or canals, whilst in Fig. 7. it is shewn that
their true planes of junction run directly along the middle of these
canals. I am however quite positive upon another point which
has been a subject of dispute among physiologists; I mean the
existence of a delicate homogeneous membrane investing this
epidermis. Such a membrane may be distinctly separated by the
action of nitric acid, from the epidermis of the corolla, filament,
and style. It is faintly marked by parallel longitudinal strie
Fig. 7, (g), and appears to coat over the whole surface of these
organs, but whether it is perforated by a fissure opposite each
stoma I did ot ascertain.

Structure of the Filament. Plate xvii. Fig. 8,9. The cellular
tissue of this organ consists of elongated rhomboidal dodecahe-
drons, as the elongated hexagons seen in its longitudinal section
sufficiently explain (Fig. 9.). -A single bundle of trachez runs up
the middle of it, invested by the peculiarly delicate fibrous tissue
already noticed. ;

Structure of the Anthers. Plate xvit. Fig. 10—12. The fibrous
cells* composing the inner coat of the anther, appeared to me
quite as distinct and perfect in the hybrid as in the parents.
Nor did I observe the slightest difference in the formation and

* See Purkinje “ De cellulis antherarum fibrosis, &c, 4to. Vratislavie 1830.”

of a Hybrid Digitalis. 273

condition of any part of this organ in either of the three plants.
In general, a transverse section shewed the fibrous-cells to be
arranged in a triple tier (Fig. 10.). These curious vessels seemed
to be set, as it were, upon the sides and edges of void
dodecahedral and other polyhedral spaces, as though certain
original cells of these shapes had disappeared and left this frame-
work of their structure alone standing. The triple tier is not
distinguishable upon looking directly down upon the inner surface
of the anther (Fig. 11.), but some of the fibrous-cells may be seen
standing upon the junction-edges of the cells of the epidermis,
where this membrane has been partially cleaned of the inner
coating composed of them. Fig. 12. (2) is the appearance which
they assume when detached by digestion in nitric acid: (k) bemg
the cells of the epidermis, (Z) an accidental appearance in a
grain of pollen recalling somewhat of the character of the grain
figured at Plate xvr. Fig. 3. H.

Structure of the Ovules. Plate xvit. Fig. 13. When the
corolla is expanded, the ovules are entirely composed of a con-
geries of large vesicles, and their surface has a very remarkable
and granulated appearance. At this period of their existence
I was unable to detect any thing very precise respecting the
distinction and distribution of their several parts. The fora-
men (m) however was evidently seated near the hilum, and a
darker spot indicated the chalaze (x) to be at the opposite ex-
tremity (see also Plate xvi. Figs. 1. and 4.) In the ovules of
purpurea and lutea, there is no difficulty in tracing the separate
parts of the ordinary structure, if they be examined shortly after
their impregnation ; but before their fertility is secured I have not
hitherto been able to detect in these plants, more than in the
hybrid, any thing but a homogeneous mass of cellular tissue.

Vol. IV. Part II. Mm

274 Proressor Henstow’s Examination

Possibly I have not given this part of the investigation sufficient
attention. When the ovules are digested in nitric acid, the de-
tached cells assume an oval shape, Fig. 15. (0), and are yellowish.
But among them I several times observed a larger cell (p) which
was more transparent and whiter, and which I fancied might be
the origim’ of the embryonic sack. These component parts are
best exhibited by crushing the ovule between two flat pieces of,
glass. Fig. 14. represents a monstrosity in which an ovule was
observed to stand upon a sort of pedicel.
Recapitulation. So far then as these researches have hitherto
proceeded in comparing the internal structure of the floral organs
of the hybrid with those of its parents, no appreciable difference
has been detected. The elementary vesicles of which their cel-
lular tissue is constructed seem to be all of the same size, and
consequently it is evident that fewer of these vesicles must be
employed in the conformation of any of the parts of hybrida,
and still fewer in those of lutea, than in completing the corres-
ponding parts of purpurea. But there appears to be nothing
actually defective in any part of these organs in the hybrid,
nothing wanting of whatever is to be found in those of the two
parents. ‘The nutritive apparatus more especially, so far as we
have examined it, seems to be quite perfect, and as the functions
performed by it in all three plants are precisely the same up to
the period when the flower falls, there seems to be no reason for
suspecting the hybrid to differ in any particular from its parents
in the perfection of its conservative organs. Since however the
functions of the reproductive apparatus appear to cease in
the hybrid before they do in the parents, it should seem that
there must be some deficiency in this part of its organization,
though it has not yet been noticed. Should the Society con-

of a Hybrid Digitalis. 275

sider the details of this examination worthy their attention, I
propose to myself the further satisfaction of prosecuting it afresh
next summer, if another opportunity should be permitted me.
Indeed I ought to add, that in the present state of this enquiry,
so little additional light has been thrown upon the great questions
connected with the phenomenon of hybridity, that I should hardly
have felt myself justified in presenting these remarks to their
notice, were it not in the hope that they might save some time
and trouble to whomsoever may be inclined to take up the sub-
ject, and possess the means of carrying on the investigation of
it still further.

MM 2

276 Proressor HeNstow’s Examination

- DESCRIPTION OF THE PLATES.

PLATE XV.

Tue raceme (Fig. 1.) and radical leaf (Fig. 2.) of the Hybrid.

PiatEe XVI.

The various parts of the floral organs in the three plants contrasted together. The

is)

details are at pages 4 and 5.

As the same parts in the three columns are marked by corresponding letters in
three alphabets, viz. in Roman capital, small Italic, and Greek characters, it
will be unnecessary to refer to more than the figures in one compartment for
the purpose of explaining those in the others.

Calyx. A. sepals separated and spread open: B. their marginal hairs magnified :
ec. with supernumerary sepal: d. to J., arrangement of the sepals during in-
florescence. N.B. These sections do not refer to the arrangement of the
sepals in estivation, which by some neglect I omitted to notice.

Corolla.

. with supernumerary petal: ry. ditto blended with the tube and forming a
notched lower lip.

Male Organs. A. is of the natural size; the rest are more or less magnified.

A. Position of the stamens in the tube of the corolla: B. a front, and C. a back
view of the anthers: D. dry, and E. moistened grains of pollen, lying on
squares representing the ;4, of an inch: F. a grain exploding upon the ap-
plication of moisture: G. three grains taken from off the surface of a withering
stigma, with their tubes (boyaww) exserted: H. a grain with three lighter

blotches on the surface.
Female Organs. A. is of the natural size; the rest are more or less magnified.

A. pistil: B. stigma: C. transverse section of the ovarium: D. an ovule at the
period of the flowers expansion, placed on a micrometer divided to the 3; of
an inch.

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of a Hybrid Digitalis. 24

PuatTe XVII.

Anatomy of the parts of fructification. All the figures excepting Fig. 4, are more
or less magnified. The same letter is always employed to designate the same
parts in the different figures.

Fig. 1, 2. Ovarium, longitudinally divided; in the first case perpendicular to, and
in the second down the plane of the dissepiment.

a. The pedicel with its circle of vascular bundles surrounding the axis: 6, branches
of this circle given off to the calyx and corolla: c, two larger bundles which
run up the pericarp, along the future line of its dehiscence, and rise through the
whole length of the style: d, separation of the vascular bundles into two bands
which enter the two lobes of the placenta near their base, and rising through
their substance d', again separate and subdivide, giving off single vessels to
the bases of the ovules: e, the stigmatic tissue descending through the style
to the summit of the placenta.

Fig. 3. A transverse section through the summit of the oyarium, and again through
the base of the style. The valve and ovules of one cell are removed. The letters
designate the same parts as in the last Figure, with the additions of ec’, small vascu-
lar bundles rising through the pericarp, all of them terminating in the base of the
style.

Fig. 4. Ripened pericarp of the Hybrid, of the natural size.

Fig. 5. The same magnified, with one valve removed—exhibiting the dissepiment, and
one lobe of the placenta, which is still fleshy, and covered by abortive ovules:
e a cavity left by the drying up of the stigmatic tissue.

Fig. 6. Epidermis of the corolla, with a glandular hair and two stomata (/).

Fig. 7. The same digested in nitric acid and more highly magnified; g, being the
investing pellicle faintly but very regularly striated.

Fig. 8, 9. Filament ; transverse and longitudinal sections.

Fig. 10. Anther; a section perpendicular to its coats, exhibiting the triple tier of its
fibrous cells.

Fig. 11. A fragment of the coats of the anther viewed on the inside perpendicularly
to its surface, which is partly divested of the fibrous-cells.

Fig. 12. Details of the anther after it has been digested in nitric acid; h, fibrous-cells
k, vesicles of the epidermis; /, a grain of pollen peculiarly marked.

278 Proressor Henstow’s Examination, &e.

Fig. 13. Ovule; m, foramen; 7, chalaze.
Fig. 14. Monstrosity of ditto.

Fig. 15. Details of the ovule after digestion in nitric acid; 0, the smaller vesicles com-
posing the bulk of the ovule; p, a paler colored vesicle occasionally found among
the former.

Pruate XVIII.

Highly magnified sections of the style and ovariuwm. Wherever the same letters are
used in this plate as in the last they designate the same parts.

Fig. 1. Transverse section of one quarter of the upper part of the ovarium. Fig. 2.
Transverse section of the style. Fig. 3. Longitudinal section of the stigma, and
part of the style. Fig. 4. Longitudinal section of the base of the style and apex
of the ovarium, perpendicular to the plane of the dissepiment.

c, the two large veins, or bundles of trachew, which rise through the whole length of
the style: c', the numerous smaller veins which terminate in its base: d’, frag-
ments of the vascular bundles which rise into the placenta and branch off to the
ovules: e, stigmatic tissue descending down the centre of the style to the summit
of the placenta; e', the same tissue coating over the swrface of the placenta, and
passing round the bases of the ovules: m, foramen, and n, chalaze, indicated by
darker spots; and in Fig. 1. the position of a raphe is apparent through the ovule,
by a darker band extending from the hilum to the chalaze: q, vesicle of the
stigma: 7, tubular vesicles of the stigmatic tissue.

Fig. 5. 0, Vesicles of the cellular tissue of the style detached by digestion in nitric
acid: p, one of them marked by a transverse band, when seen more highly
magnified.

IX. On a remarkable Modification of Newton's Rings.

By G. B. AIRY, M.A. F.R. Ast. Soc. F.G.S.

PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY,
LATE FELLOW OF TRINITY COLLEGE, AND FELLOW OF THE
CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read Nov. 14, 1831.]

Tue following variation of the observation of Newton’s coloured
rings will it is hoped be considered as conclusive, so far as it
goes, in favour of the undulatory theory of light. It was sug-
gested by a consideration of the values assumed in_ particular
cases by Fresnel’s general formula for the intensity of reflected
light; experiment has entirely confirmed my anticipations: and
the fact appears to be perfectly inexplicable on any theory of
emissions.

To begin with a case generally known, suppose that a convex
lens of great focal length is placed on another convex lens, or
on a plane glass, or on a concave glass where the radius of con-
cavity is greater than the radius of the convexity which rests
upon it: and suppose common light to fall on it, and to be received
by the eye after reflection. A set of rings is seen with a remark-
ably black spot in the center: considerable pressure being some-
times necessary to insure the blackness of the central spot. On
inclining the incident ray, the rings dilate, but the center remains
perfectly black, and continues so till the direction of the incident
light is parallel to the upper surface of the lens.

280 Proressor Airy on a remarkable

Now instead of common light, suppose that polarized light is
incident. The general appearances are not altered; the only
modification being that, if the plane of polarization is perpen-
dicular to the plane of reflection, when the angle of incidence
becomes equal to the polarizing angle the whole of the reflected
light disappears, and with it the whole system of rings. But on
increasing the angle of incidence the reflected light again appears,
and the system of rings is restored, with the center black as
before. It is indifferent whether the light is polarized before
incidence, or a polarizing substance (as a plate of tourmaline) is
placed between the lenses and the eye.

Instead of placing the lens on a plate or lens of glass, let
it be placed on a polished plate of metal. If common light is
used, the rings are seen as before (though not so black), but the
central spot is perhaps not quite so large as before. On increasing
the angle of incidence up to 90°, the rings dilate as before, but
the central spot, I think, diminishes a little (at least in proportion
to that formed when the lens is placed on a glass plate). In
this case then, the general character of the appearances is almost
exactly the same as before.

If however the lens is placed on a polished plate of metal,
and if the light is polarized either before or after incidence* in

* | have carefully verified this assertion (that it is indifferent whether the light is polarized
before or after reflection) because I think that it leads to important theoretical conclusions.
If polarization were a modification of light (as Dr. Brewster and others have supposed), it
might be conceived that polarization before incidence might destroy its power of producing
rings at a certain angle, or might change the tints; but when the reflexion is performed,
and the rings are actually visible to the eye with a dark center, it seems quite inconceiv-
able that any modification or physical change in the light should make that center appear
white. The satisfactory explanation is, that polarization is a resolution of the vibrations into
two sets at right angles to each other, performed in such a manner that the two sets can
in general be separately exhibited, and that in this instance only one is transmitted to the eye.

Modification of Newton’s Rings. 281

a plane perpendicular to the plane of incidence, the phenomena
undergo a remarkable change. When the angle of incidence is
small, the central spot is dark, but not black. It continues
dark and without any sensible alteration of size as the angle
of incidence approaches to the polarizing angle of the glass.
The rings become faint (the central dark spot retaining the same
magnitude as long as it is visible) and disappear* when the
angle of incidence is equal to the polarizing angle. On increas-
ing the angle of incidence by a very small quantity the rings
are again seen, of the same magnitude, but the central spot
without any sensible alteration of size is now white. And the
intensity and colour of the rings appear to be, in every part,
complementary to what they were before. This state continues
with very little alteration in the magnitude of the central white
spot, till the angle of incidence becomes 90°.

If the lhght is polarized before or after incidence in the plane
of reflexion, there is no such change of colours. The magnitude
of the central spot is altered; but through all variations of the
angle of incidence the central spot still remains darker than the
ring which immediately incloses it.

If common light is incident at an angle greater than the
polarizing angle, and a plate of tourmaline is held between the
glass and the eye, the axis of the tourmaline being in the plane
of reflection, the central spot is black. On turning it to the right
or the left, the dark spot dilates and a white spot arises in the
center, which acquires its maximum diameter when the axis of
the tourmaline is perpendicular to the plane of reflexion. On

* This simple fact (the disappearance of the rings while abundance of light is re-
flected from the metal) seems to be satisfactory evidence, if any were wanted, to shew
that the rings are produced by interference only.

Vol. IV. Part II. Nw

282 Proressor Airy on a remarkable

turning farther, the white spot contracts and vanishes, and the
dark spot diminishes till the axis of the tourmaline is again in
the plane of reflexion.

The plate cf metal used in these experiments was a telescope
speculum, with tolerably bright surface. To guard against all
sourees of error from want of contact of the lens and the metal,
a broad flat ring of thick sheet lead was placed on the lens,
and was sometimes loaded with a weight of about six pounds,
distributed over its circumference. The changes in question
therefore had no connexion with the changes of colour in the
central spot when the lens and glass plate are not perfectly in
contact. And it will easily be seen that they have nothing m
cemmon with the change from black to white produced in some
of Sir W. Herschel's experiments (Phil. Trans. 1807 and 1809),
the latter being merely a substitution, by a kind of slight of
hand, of the rmgs of transmitted light for those of reflected light.

The explanation of this phenomenon will be found in the
expression for the intensity of rings produced by the interference
of two streams of reflected light, the quantity of light reflected
being expressed by Fresnel’s formula. If + be the angle of inci-
dence within the glass and ’ the corresponding angle of refraction,
and if the light be polarized in the plane perpendicular to the
plane of reflexion (that is, if the vibrations of the particles of
ether take place wholly in the plane of reflexion), and if the
extent of displacement in the vibrations before incidence be re-

a ; 5
presented by a.sin —~ (vt — 2), then that in the vibrations after

reflexion from the second surface of the glass will be

tan (« — v)
eae ope = =(vt— x).

Modification of Newton’s Rings. 283

This stream of light interferes with that reflected from the surface
of the metallic reflector: in which the expression for the displace-

ment is a.A.sin < (vt-2'— B). In this formula 2 differs from «

by the difference of paths described by the two rays, or rather by
the space in air equivalent to that difference ; which, if 7 be the
thickness of the plate of air, is 27’ cos /. Thus the displacement

. . . . 9 ¢
by the metallic reflexion is aA.sin = (vt—2—2 T cos '— B), where

A and B are probably functions of “ whose form is unknown.
We can assert however that when / is small, 4 is positive and
B is not great (else, as will be seen from the subsequent ex-
pressions, there would be a white spot at the center): and the
gradual change of appearances, except at the polarizing angle,
makes it probable that B is always small. At any rate there
is not the slightest reason to believe that 4 or B, which depend
only on the properties of the metal, would undergo any sudden
change exactly at the polarizing angle of the glass.

Now the peculiar phenomenon which is the principal subject
of this paper is thus explained. The central spot, where 7'=0,
is produced by the composition of two displacements

tan (c—’) . Qa
Pani HL) sin > (vt—2)

and a.A.sin =e (vt-.x) (considering B=0).

As « is greater than «, the former of these is negative when .
is small, and the latter is positive. Consequently these displace-
ments partly or entirely destroy each other, and the center is
dark. But at the polarizing angle, «+: (by Brewster’s law) =90°,

and tan (+) is infinite; the first expression vanishes, and the
NN 2

284 Prorrssor Airy on a remarkable

only light that comes to the eye is that reflected from the metal,
unmixed with the other light. Beyond the polarizing angle «+:

tan (\—c’) .. Se alg
moins positive: the two

is >90°, tan(:+/) is negative, and a
displacements have the same sign, and therefore are added toge-
ther, and the central spot is therefore bright.

To shew distinctly. that the character of the rings as well as
that of the central spot ought to change at the same time, it will
be best to take the general expression. We have to add together

tan («<—v’)

“tan (¢+0) sin 5" (ot - e)

and a.A.sin = (vt—a2—2T cos — B).

The sum is

i, Wu («—¢)

27 —— SI .... 27
Tee Wie cos 57 3 T cos + Bl sin > (vt—2)

—aA.sin a .2T cos (+ B. cos 22 (vt—2).
If this be put in the form
P sin a (vt—x—Q),

it is easily seen that

tan (c—¢’)

aie. (aes

Bip > eS
+A cos" 27 cos +B + 0°A*. sin' "2 T c08 4B

=a \ae eo +4°+2A. cos = 2T cs (+B. aaeeaitl
tan® (c +¢’) tan (+)
Now P* represents the intensity of the light in the mixture:
consequently the last formula will represent that intensity, or
the intensity of light that comes to the eye. It is easily seen

Modification of Newton’s Rings. 285

that if « and © are small, « being <, this is minimum when
2T cos ( + B=0, or =X, or =2d, &c. and maximum when

7 Xr
2T cost +B=5) or = —, &e.:

and that this continues till «+¢=90°: but as soon as .+¢ exceeds
90°, the expression is maximum when 2T cos + B=0, or =X, or
= 2), &c., and minimum when
2T cos’ + B= x, or = ae &e.

If B be small, the first of these represents rings similar to
Newton’s reflected rings, and the second represents rings similar
to his transmitted rings.

If the light is polarized in the plane of reflection, the ex-
pression for the displacement in light reflected from the lower
sin («—v)

—— i oO ge sig
ain G0)” which does not change sign

surface of the glass is a.

like the former: and a similar train of reasoning. shews that the
character of the rings which it produces will not suddenly change.

If the light is polarized in a plane inclined by the angle
a to the plane of reflection, we must regard this as consisting
of two vibrations, one perpendicular and one parallel to that
plane, whose proportions are cosa: sina. The original vibration
perpendicular to that plane must now be taken

a@.cos a.sin n= (vt—2),
and that parallel to the plane

° Abie
a.sin a.sin ao (vt—x).

286 Proressor Airy on a remarkable
The intensities of the reflected streams are found to be:

For the light polarized in the plane of reflexion,

a cos” + fmt) 4 Ato. cos = 2T cos ' + B. anu eal
sin® (« +’) sin (¢ +c’)
(where 4’ and B’ are quantities analogous to 4 and B, and
where it is probable that A’ is always positive and B not very
great).

For the light polarized perpendicular to the plane of reflexion,

a? cin’ « f= es + 4424008 2% BP eos + BSB) al
The intensity of the compound light which comes to the eye is
the sum of these. The value of 7 corresponding to the middle
of a bright or dark ring is determined by making the sum a
maximum or minimum with respect to the variation of 7. This
gives,

tan = (27 cos ¢ te hs

A’ cos? a. cos («—’)— A sin? a. cos (¢ +4 )

5 Predhe ooUET) F LEE oom T Ny ih)

If tan = (B-B’) is positive, and -cos (:—’’) negative; upon in-

creasing « from 0 to 90° the value of tan = (27 cos spe +)

increases to infinity and then becomes negative and decreases :

2 Bee Vis
that is oa (27 cos + 5 ) increases: and therefore the diameter

of the ring increases. Upon increasing a from 90° to 180° the op-
posite change takes place. This represents exactly the fact of

Modification of Newton’s Rings. 287

observation ; and it proves therefore that B is > B’*; a conclusion
of some inportance, as shewing that plane-polarized light reflected
at a metallic surface becomes elliptically polarized, and as con-
necting these phenomena with those of a very different kind
discovered by Dr. Brewster. It appears here that the phases of
vibrations in the plane of reflection are more retarded, than those
perpendicular to that plane.

If cos (+¢) is positive, that is, if the angle of incidence is less
than the polarizing angle, it appears that upon increasing a the
diameters of the rings ought to decrease. But it is easily seen
that the change ought to be much less than in the former case.
oe is

meas i 2
For im the former case, if one value of — (27 cos . -

: : 2 , B+B
ur+3, when a=0, then, upon increasing a to 90°, 25 (2 T cos¢ + 3 )
changes through x7+90° to ~z+180°—, and the whole change is

changes from 27+ through xz to »xr—£, and the whole change
is therefore 28, which is exactly supplemental to the former whole
change. Now in Newton’s rings formed between two lenses, 6
is 0; from the general similarity of the rings formed by a metal
reflector when a= 0, it is certain that 6 is small; consequently
though the dilatation of the rings in the former case depending

therefore 180° — 28. .But in the latter case “2 (27' cos +

: : i HE BS: .
on the increase 180°—2£ in “2 (27 cos. ae ) is considerable,

2
the contraction in the latter case (depending on the decrease 2/)

* If B were < B’, the rings would contract instead of expanding; and if B were = B’,
the diameters of the rings would not alter, but their intensity would diminish to 0, when
tan? a = — 2 eet) and rings of the opposite character would then appear.

A’ cos (+0)

288 Proressor Airy on a remarkable Modification, §c.

is small. I have submitted this to experiment, but I have not
been able to discover with certainty any alteration in the size of
the rings.

The reasoning upon which the principal experiment described
in this paper was anticipated, may probably be applied in many
similar cases. The following is a very remarkable instance. If
a lens of a low-refracting substance be placed on a plate of a
highly-refracting substance, or vice versa, and if they be illumi-
nated with light polarized perpendicular to the plane of incidence,
I expect that while the angle of incidence is less than the pola-
rizing angle of the low-refracting substance, the central spot will
be dark ; when the angle of incidence is greater than the polarizing
angle of this substance and less than that of the other, the central
spot will be bright; and when the angle of incidence is greater
than the polarizing angle of the highly-refracting substance, the
central spot will again be dark. I have not yet procured any
substances proper for the verification of this conclusion.

G. B. AIRY.

OBSERVATORY,
June 21, 1831.

X. A Monograph on the British species of Cyclas
and Pisidium.

By tHe Rev. LEONARD JENYNS, M.A. F.L.S.

AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read Nov. 28, 1831.]

Tue genus Cyclas of Bruguiere was instituted for the reception
of certain species of ‘“‘Bivalve Mollusca” inhabiting fresh water,
which were associated by the older Linnzan authors, either with
Cardium or Tellina. Only three of these shells appear to have
been distinctly known to Montagu as natives of this country,
-who referred them in his “‘ Testacea Britannica” to the former
of the two genera just mentioned. Other indigenous species have
been discovered in later years, some of which have been de-
scribed, and from time to time been made known to the public;
nevertheless, from want of having their characters accurately
defined, and still more from not being illustrated by figures suf-
ficiently large to convey a correct idea of their relative proportions,
it is not always very easy to identify them, nor to determine
how far such species are really distinct from one another, or
from those before known. It may be added also, that the syno-
nyms have been much confused, and the same name has been
applied in several instances to more than one species. It is on
these grounds that I have been induced to draw up the following

Vol. 1V. Part II. Oo

290 Mr Jenyns’s Monograph on the

paper. Having for many years paid considerable attention to
this family, during which time I have not only increased the
list of British species, but have also endeavoured to ascertain the
extent of variation to which each of them is subject; haying
likewise been fortunate in obtaining authentic specimens of many
of those alluded to by British authors, I thought that it might
be rendering a service to the conchologists of this country, if
I were to throw together, in the form of a monograph, the
observations which I had made, and to draw up an amended
list of all the species hitherto detected in this island, accompanied
by full descriptions, and illustrated by magnified figures. In my
attempt to do this, I have not merely considered the general
characters of the shell, but in determiming the species have de-
rived much assistance from attending to the animal inhabitant.
The species themselves are found in rivers, ditches, and streams ;
the smaller ones not unfrequently in the gullies that are cut in
pastures for the purpose of draining the soil: they all however
live readily in confinement for several days when kept in water;
and under these circumstances their different habits may be con-
veniently observed. Occasionally it will be found that they be-
come languid and inert, especially if they have been confined
a long time; but they may generally be roused into activity by
the sudden application of cold spring water, and this is by far
the best, method of getting a sight of the siphonal ‘tubes, which
in some cases afford goud distinguishing characters. Indeed it is
absolutely requisite to caution conchologists against drawing any
conclusions with respect to the specific distinction of these animals
from a mere inspection of the shell alone. This is so liable to
vary from age, peculiarity of situation, and probably from other
causes, that it becomes necessary in some cases to compare a

British species of Cyclas and Pisidium. 291

large number of specimens, collected from different sources, in
order to determine the characters of a single species with any
degree of precision. Occasionally the shell becomes exceedingly
ventricose at the expense of its height, which is thereby consider-
ably diminished; and the valves which perhaps naturally meet
at an acute angle, under such circumstances meet at an obtuse
one. This is particularly the case with one or two varieties here-
after to be described. Neither can sculpture be relied upon, the
strie varying exceedingly in number and distinctness according
to the nature of the water in which the shell is found: a circum-
stance of which Dr Leach was not sufficiently aware when he
formed three species out of Pisidium amnicum. Age likewise
produces great changes: not only are young shells much more
compressed than adult ones, but in many instances the relative
proportions of their parts are different. Indeed in the case of
the minute species, so great and general a similarity prevails
amongst their young, that it is hardly possible to identify them
in this state without the closest examination.

After what has been stated, I need hardly add, how cautious
I have been in characterizing the minuter species of this group,
and that it is not till after repeated observations upon very ex-
tensive series of each, including many varieties from different
localities, that I venture to bring forward the following list of
such as are found in this country, as one which I trust will
be found more complete than any which has appeared hither-
to.

It will be observed, that with respect to the arrangement of
these shells, I have deviated from that of Lamarck and most
authors in referring them to two genera. This I have done in
conformity with the views of Pfeiffer, who in his excellent work

002

292 Mr Jenyns’s Monograph on the

on the land and fresh water Mollusca of Germany*, first insti-
tuted the genus Pisidium for the reception of those species which
are characterized by an ineequilateral shell, and a single siphon
at the posterior extremity of the cloak. This distinction, founded
upon the structure of the animal as well as that of the shell,
appears to be perfectly natural; while there is little doubt that
its utility will be soon acknowledged, when increased attention
shall have been paid to this family, and future discovery have
still further augmented the number of species, which we have
every reason to expect will be the case. The propriety, however,
of such a division of the old genus Cyclas will be best seen from
the following comparative view of the characters of those adopted
in this paper.

Gen. I. CYCLAS, Lam. Pfeiff.

Animal: Pallium anticé, pro pede exserendo, apertum, posticé con-
natum, et in tubum siphonalem longum, duplicem, contractilem, exten-

sum. Pes linguiformis, valde extensilis.

Testa corticata, suborbicularis, subequilateralis. Dentes cardinales
minuti; in dextra valvula unicus plus minusve complex; in sinistra
duo obliqué collocati. Dentes laterales longitudinales, compressi, lamel-
liformes, in valvula dextra subduplicati. Ligamentum externum, posti-

cum, lateri longiori insitum.

Gen. II. PISIDIUM, Pfeiff.
Animal: Pallium anticé, pro pede exserendo, apertum, posticé con-
natum, et in tubum siphonalem brevem, simplicem, contractilem, ex-

tensum. Pes linguiformis, valde extensilis.

* Systematische Anordung und Beschreibung deutscher Land-und Wasser-Schnecken, 5:c. von
Carl Pfeiffer. Cassel, 1821.

British species of Cyclas and Pisidium. 293

‘Testa corticata, subovalis, ineequilateralis. Dentes cardinales minuti;
in dextra valvula unicus plus minusve complex; in sinistra plerumque
duo. Dentes laterales \ongitudinales, compressi, lamelliformes, in valvula

dextra duplicati. Ligamentum externum, posticum, lateri breviori insi-

tum.

Although Pfeiffer has the merit of having first separated the
above genera, his characters are not clearly defined, from the
circumstance of his having confounded the anterior and posterior
parts of the shell. The extremity from which the siphonal tube
is protruded, and which strictly speaking is the posterior, he has
termed differently in his descriptions of the two genera. I have
adopted the views of Blainville with reference to this point, as
well as in the selection of terms employed in characterizing the
species, and accordingly it is to his “Manuel de Malacologie”
that I must refer for an explanation of all such as occur in the
following paper. The best characters for distinguishing the above
genera are drawn from the structure of the siphonal tube, and the
position of the hinge considered relatively to the two extremities
of the shell. In the genus Cyclas, the tube is capable of protru-
sion to a considerable Jength, and although single at the base, is
always divided at the apex, the upper portion, which is generally
the shorter of the two, acting as the anus, while the lower, which
is both longer and has a larger aperture, serves to conduct the
water to the branchiz. In Pisidium, this tube is single and un-
divided throughout its whole length, and although to a certain
degree admitting of extension and contraction, is never protruded
to the distance that it is in Cyclas. Indeed in both genera it
appears to be quite a matter of pleasure whether it be exserted
at all; the animal being often seen, both at rest and in motion,

294 Mr Jenyns’s Monograph on the

with the tube either entirely concealed within the shell, or ex-
tended to its utmost limit. Another obvious distinction between
these two genera, is afforded by the position of the hinge and
cardinal teeth, with respect to the extremities of the shell. The
teeth themselves are similar in the two instances, at least in
general character; although subject to minute variation of form,
more especially the cardinal tooth in the right valve, even
amongst individuals of the same species; but the situation of
the hinge is essentially different in Cyclas and Pisidium. In the
former genus it is nearly central, the posterior portion of the
shell being only to a slight degree longer than the anterior: in
the latter, it is placed towards one end, and the anterior portion
is obviously the longest; and although the excess of length in
the first section of this genus is not very considerable, yet it will
always be found in front of the hinge and not behind it. This
will be made to appear more clearly by inspecting Plate x1x. in
which Fig. 3. represents the hinge of Cyclas calyculata, and
Fig. 4. that of Pisidium amnicum. In each case the right and
left valves are distinguished by the letters 4 and B respectively,
whilst @ points out the relative position of the cardinal teeth.

Having made the above observations upon the generic
distinctions afforded by these shells, I shall now proceed to
characterize the species themselves in the order of their respective
affinities.

Gen. I. CYCLAS.
Sp.1. C. rivicola, Leach.

C, testa globoso-ovali, ventricosa, striata; umbonibus obtusis; anticé
lunulé flava impressa ; ligamento cardinali conpicuo.

Long. 103 lin. Alt. 84 lin. Crass. 64 lin.

British species of Cyclas and Pisidium. 295

Cyclas rivicola, Leach MSS.—Lam. An. sans. Verteb. 5. 558.—Pfeiff.

Land-und Wasserschn. 121. t. 5. J. 3-5.—Turt. Conchyl. Brit. 248.
“# U1. f. 13.—Turt. Man. of Brit. Land and Freshw. Shells. 12.
J. 1.—Flem. Brit. An. 452.

Tellina cornea B, Mat. and Rack. Linn, Frans. 8. 59.—Turt. Conch.
Dict. 180.

Cyclas cornea, Draparn. Hist. des Moll. 128. t. 10. f. 1—3.—Brard,
Hist. des Cog. 219. t. 8. f. 2. 3.

Jun. C. xquata, Shepp. MSS. Brit. Mus.

Animal mihi ignotum.

Testa globosa, subovalis, ventricosa, solidiuscula, eleganter et distincteé
striata; fusco-virescens, fasciis 2—3 saturatioribus, margine basali
luteo; intus cerulescens: umbones tumidi, pallidiores, linea nigricante
plerumque circumscripti: margo dorsalis anticé lunula, posticé fissura
distincté impressé, utréque flavescenti: ligamentum, cardinale con-
spicuum. R
This species is at once distinguished from the next, and from all

the other British ones of this family by the superior size of the shell:
the animal I have not seen. It appears to be confined to rivers, and is
found I believe abundantly in the Thames, as well as in some other
parts of the country. The largest specimens in my possession are from
the Trent in Nottinghamshire. In its young state, it appears to be
identical with the C. equata of the Rev. R. Sheppard, according to
specimens so named by that gentleman in the British Museum.

Sp. 2. C. cornea, Lamark.
C. testa suborbiculari, globosa, tenerrimé striata; umbonibus obtusis ;

ligamento cardinali inconpicuo.

Long. 64 lin, Alt. 5. lin. Crass. 4 lin.

296 Mr Jenyns’s Monograph on the

Cyclas cornea, Lamarck, 5. 558.— Pfeiffer, 120. t. 5. f. 1, 2.— Nilsson,
Hist. Mollusc. Suec. 96.—Turt. Conch. Brit. 248. t. 11. f. 14.—Turt.
Man. 13. f. 2.—Fleming, 452.

Tellina cornea, Linn. Syst. Nat. 1. 1120.—Gmel. 3241.—Linn. Trans.
8. 59.—Dillw. Cat. of Shells, 1. 104.—Don. Brit. Shells, t. 96.

Tellina rivalis, Mud/l. Verm. Hist. 2. 202.
Cardium corneum, Mont. Test. Brit. 86.
Cyclas rivalis, Draparn. 129. t. 10. f: 4, 5—Brard, 222. t. 8. f. 4, 5.
Var. p.
Testa subglobosa, versus marginem basalem complanata ; umbonibus
tumidis, pellucidis, valde prominentibus.

Long. 54 lin. alt. 43. Crass. 34 lin.

Cyclas stagnicola, Leach MSS. Brit. Mus.
Tellina stagnicola, Sheppard, Linn. Trans. 14. 150.

Animal album, viviparum: tubi siphonales subelongati, carneo_pallide
colorati; superiore subconico, apertura parva apice perforato; infe-
riore cylindraceo, truncato, apertura ampliori: pes testam longitudine

paulo superans,

Testa globosa, suborbicularis, ventricosa, tenuis, levissimé striata; nunc
virescenti-fusca, zonis 1—3 lutescentibus, quarum 1 plerumque mar-
ginalis latior; nunc omnind fuscescens aut lutescens: umbones obtusi;
in var. @ valde prominuli, quasi inflati, pellucidi: lunula vix ulla:
margo dorsalis posticé litura nigricanti duplici sepe notatus: liga-
mentum cardinale inconspicuum.

This very common species is a general inhabitant of rivers, ponds,

and ditches throughout the country, and appears to thrive equally well
both in running and in stagnant water. In confinement it will oc-

British species of Cyclas and Pisidium. 297

casionally ascend the sides of the yessel in which it is kept, and
during locomotion I have observed that the tubes are either partially
exserted, or entirely concealed.

The variety @ agrees with the series of specimens in the British
Museum named by Dr Leach C. stagnicola. I am inclined to believe
also that it is the same with the Tedlina stagnicola of Mr Sheppard,
as the remarks made by this latter gentleman with respect to the
peculiar appearance of the umbones, apply very exactly; and it is
particularly stated that he first received specimens from Dr Leach
under the above name. Nevertheless I feel satisfied that it is a mere
variety of C. cornea, as the animal js exactly the same in the two in-
stances; and with regard to the peculiarity of the shell, many interme-
diate specimens may occasionally be met with. It is necessary, however,
to mention, that the name of stagnicola appears to have been applied
by Dr Leach at different times to two distinct species. The series of
shells at present so named in the British Museum, are certainly the
variety of C. cornea now under consideration; but I possess two spe-
cimens of a shell which came originally from Dr Leach, and which
have the name of stagnicola under them in that gentleman’s own hand-
writing, evidently belonging to the C. calyculata of Draparnaud, (Var. Y-
of this paper,) and I am inclined to think that it was this latter shell
which was formerly sent by the Doctor to Lamarck under the above
name, and considered by that author to be a mere variety, not of
C. cornea, but of the species last mentioned.

Other varieties of C. cornea, besides those above-mentioned, are not
uncommon. Occasionally the shell exhibits gibbosities, and the margin
becomes very obtuse: at other times the valves are much compressed,
and their margins meet at an acute angle. In the fens of Cam-
bridgeshire, a small variety is not unfrequent in the turf pits almost
globular, and somewhat similar both in size and shape to a pea.

Vol. IV. Part II. Pp

298 Mr Jenyns’s Monograph on the

I may here observe that this, and all the other species of this
family breed readily in confinement, during the spring and summer
months, They are probably ovoviviparous; and the young appear to
remain for a certain period within the folds of the branchiz previous to
their exclusion, since many may be found of different sizes within the
parent at one and the same time. They have the faculty of producing
long before they are arrived at their full growth, and even some indi-
viduals which are themselves so immature as to possess hardly any of
the distinguishing characters of the species, frequently contain young of
a sufficient size to be seen from without through the transparent valves.

In distinguishing this and the last species, authors have frequently
drawn their essential characters from the presence and number of the
longitudinal, or as they term them ¢ransverse grooves, indicative of
the different stages of growth. But as these are very uncertain marks,
depending upon age and other circumstances, I have not thought it

necessary to notice them at all. The colour is not less variable.

Sp. 3. C. calyculata, Draparnaud.
C. testa subrhombea, compressa, tenui, albo-lutescenti, diaphana; na-
tibus prominentibus, acutiusculis, tuberculosis.
Long. 53 lin. Alt. 43 lin. Crass. vix 3 lin.
Cyclas calyculata, Draparn. 130. ¢. 10. f 13, 14.—Lamarck, 5. 559.
Pfeiffer, 122. t. 5. f. 17, 18.— Nilsson, 99 —Turt. Man. 14. f. 3.
Cardium lacustre, Montagu, 89. :
Tellina lacustris, Linn. Trans. 8. 60.—Twurt. Conch. Dict. 180.
Cyclas lacustris, Turt. Conchyl. Brit. 249. t. 11. f 18.
Var. B.—Taz. xrx. Fig. 1.
Testa orbiculato-rhombea, minus compressdé, subdiaphana, fusco-ru-

fescente.
Long. 43 lin. Alt. 4 lin. Crass. 27 lin.

British species of Cyclas and Pisidium. 299

Cyclas lacustris, Alder in Trans. Nat. Hist. Soc. Newcast. 1. 40.
Brit. Mus. MSS.

Var. y-

Testa orbiculato-rhombea, minus compressa, subdiaphana, rufescente :

natibus nigricantibus, mints prominulis.

Cyclas stagnicola, Leach (olim.)

C. calyculata, (2), Lamarck, 5. 559.

Animal (in Var. 8) album, tubis siphonalibus concoloribus; hi valde elon-
gati, nunc superiore, nunc inferiore alium longitudine superante,
quoad formam fere ut in specie preecedenti.

Testa quam maximé variabilis, rhombea, orbiculato-rrhombea, sub-ovalis,
vel exacte orbicularis; plus minusve compressa, tenuis, diaphana,
levissimé striata; plerumque cerulescenti-alba, zoné marginali lutes-
cente;—interdum fusco-rufescens, minus diaphana, apice nigricanti:
nates acutiuscule, tuberculose, in a et 8 prominentes, interdum etiam
subinflexee: ligamentum inconspicuum.

I feel satisfied that the above described shells are only varieties of
one species, and all referable to the C. calyculata of Draparnaud. One
of them is the C. lacustris of the British Museum, and also of Mr Alder,
as I have been enabled to ascertain from specimens kindly forwarded
to me by that gentleman. Var. y, which only differs from the last in
having the tubercles on the beaks not quite so prominent and well
defined, I believe to be the variety, as I have already stated, originally
sent to Lamarck by Dr Leach under the name of C. stagnicola. Mr Alder
was of opinion that his shell was the C. dacustris of Dyraparnaud, but
as Lamarck has referred the variety last mentioned (which differs so
little from it) to the present species, and as he was acquainted with
both the C. calyculata and lacustris, there can be little doubt that this
last is distinct from either of the above——Indeed I have never seen
any British shell exactly answering to the C. lacustris of the continental

300 Mr Jenyns’s Monograph on the

authors. Draparnaud appears to be the first who made a distinction
between this shell and the C. calyculata, and this distinction has been
since acknowledged not only by Lamarck, but also by Pfeiffer and
Nilsson; but all the British specimens that have fallen under my ob-
servation with the name of C. lacustris attached to them, are in my
opinion nothing more than mere varieties of the species under con-
sideration.

As Muller has described only one of these two species, I consider
it doubtful to which his description applies. I have therefore made
no reference to his work in the present instance.

C. calyculata is much less abundant in this country than C. cornea.
Montagu met with it in Devonshire and Wiltshire. Mr Alder finds
it near Newcastle, but says that it is rare; and it has occurred sparingly
to myself in two or three parts of Cambridgeshire—Var. 8, I observed
last summer (1831) in considerable abundance in one pond on Bookham-
Common in Surrey, and some which I kept by me alive for a few days,
showed more activity than the last species, readily and frequently as-
cending the sides of the vessel, and walking, like Physa Hypnorum,
on the under side of the surface of the water*. Occasionally they re-
mained in a quiescent state at the bottom with their posterior extremity
elevated, and the siphonal tubes exserted to a considerable length, often
nearly equalling that of the shell itself—Var. y. I have received from
the North of England.

In young specimens the tubercle at the apex of each valve, so charac-

teristic of this species, is relatively much larger than in the adult shell.

* This phraseology is not strictly correct, but is perhaps sufficiently intelligible. The
action intended, consists in the animal extending its foot along the surface of the water
with its shell immersed, and in an inverted position. In this manner, it contrives to
traverse the vessel from side to side as though it were crawling along a solid plane.

British species of Cyclas and Pisidiun. 301

Gen. II. PISIDIUM.

* Testa parum inequilaterali.

Sp.1. P. obtusale, Pfeiffer—Tagp. xx. Fig. 1—3.

P. testé globosa, obliqué subovali, tenuissimé striata; umbonibus pro-
minulis, obtusissimis.

Long. 12 lin. Alt. 14 lin. Crass. vix. 14 lin.

Pisidium obtusale, Pfeiffer, 125. t. 5. f. 21, 22.— Brown in Edinb.
Journ. of Nat. and Geog. Scien. 1. 413.

An Cyclas obtusalis? Lamarck, 5. 559.

Pera gibba, Leach, MSS. Brit. Mus.

Var. p.

Testa ovato-trigona, ventricosissima, margine obtusissimo.

Cyelas obtusalis, Nz/sson, 101.

Animal album; tubo siphonali abbreviato, subconico; pede valde ex-
tensili, testa dimidio et ultra longiori.

Testa globoso-ovalis, ventricosissima, crassitudine fere altitudinem zquanti,
nitida, subtiliter striata; plerumque virescenti-nigra vel ochraceo-
nigricans, zona marginali (junioribus latissima) lutescenti, interdum
subaurantia; raritis omnino lutescens: umbones tumidi, obtusé ro-
tundati, pauld prominentes.

Var. 8. gaudet testa ventricosiori, margine basali obtusissimo, quo minua-
tur altitudo, et forma magis trigona vel ovato-trigona provenit.
Hee varietas plerumque nigricans, ochraceo plus minusve fucata.
This species, which is distinguished from all its congeners by the

extreme convexity of the shell, is certainly the P. obtusale of Pfeiffer,

and probably the Cyclas obtusalis of Lamarck, but from the short de-

302 Mr Jenyns’s Monograph on the

scription given by this last author, a little doubt attaches itself to the
latter synonym.—The variety 8, also accords exactly with the Cyclas
obtusalis of Nilsson. Dr Leach called it Pera gibba, but I do not feel
certain that all the specimens on the board so named in the British
Museum are referable to this species.—It occurs not unfrequently in
Cambridgeshire, inhabiting small splashy pools and other stagnant waters,
and I have observed that it is often to be found in company with
the Physa Hypnorum Drap. It is by far the most active and lively
species that I am acquainted with, being always in motion, and _ re-
siding less at the bottom than the rest of this family. It transports
itself rapidly along the under side of the surface of the water, and
appears to delight much in floating masses of conferve and other
weeds. Dr Leach’s specimens came, I believe, from the neighbour-
hood of Battersea Fields; and I have myself also met with it in
other parts of Surrey.

Obs.—The measurements of this species are usually much less than
those above given.
Sp. 2. P. pusillum, Nobis.—Tas. xx. Fig. 4—6.

P. testa orbiculato-ovali, compressiuscula, subtilissimeé striata, vix in-

equilaterali; umbonibus parim prominulis.
Long. 1 lin. Alt. 1} lin. Crass. 1 lin.
Tellina pusilla, Twurt. Conch. Dict. 167.
Cyclas pusilla, Taurt. Conchyl. Brit. 251. ¢. 11. f 16, 17-—Turt. Man.
16. f 7.

fontinalis, Ni/sson, 101.—Draparn. 130. ¢. 10. 72 8—11 ?
gibba, Alder in Trans. Nat. Hist. Soc. Newcast. 1. 41.

Euglesa Henslowiana, Leach MSS. Brit. Mus.
Var. B.

Umbonibus magis prominentibus.

British species of Cyclas and Pisidium. 303
Var. vy.
Striis profundits incisis.
Animal album; tubo siphonali brevi, nune cylindraceo, nunc subconico,

margine integerrimo; pede testam longitudine pauld superante.

Testa variabilis, plerumque orbiculato-ovalis, interdum suboblonga mar-
gine dorsali recto, vix inaquilateralis; pracedenti multd magis com-
pressa, marginibus acutis ; seepius extranea rubigine obtecta, qué remota,
apparent striz subtilissime, non nisi oculo armato conspiciende; in
var. ry. nitida, striis distinctis, profundits incisis: umbones subdepressi,

parum prominuli, interdum subacuti.

This species appears to be the Cyclas pusilla of Turton, whose
description and figure in his “British Bivalves,” apply with tolerable
exactness. Specimens also of that shell which I received some time
since from the Rev. R. T. Lowe, with the assurance that they were
authentic specimens originally from Dr Turton himself, agree with mine
in every essential particular, although more compressed, and with the
umbones not quite so obtuse and prominent. Nevertheless, I am inclined
to think that this name has been occasionally applied to more than
one species, particularly to some of the varieties of P. pulchellum here-
after to be described, which I have received from one or two collectors
as the shell above-mentioned. I likewise consider this species as sy-
nonymous with the Cyclas fontinalis of Nilsson, although I entertain
some doubts as to its identity with the C. fontinalis of other conti-
nental authors. Draparnaud especially has comprised under this name
two varieties differing so materially in size, as to render it hardly pro-
bable that they belong to the same species.

Pisidium pusillum is distinguished from P. obtusale by the shell being
much more compressed than in that species, and by the margins of

the valves meeting at an acuter angle: the hinge is also nearly central.

304 Mr Jenyns’s Monograph on the

the anterior extremity still being in a slight degree longer than the
posterior. It is by no means of uncommon occurrence, residing chiefly
at the bottoms of drains and ditches, where I have often found it
buried at a considerable depth in the soft mud. It appears to be some-
what amphibious in its habits. Nilsson observes that it is often to be
met with between the bark and the wood of decayed timber in wet
places, and I have myself noticed that in confinement it will frequently
leave the bottom of the vessel, and ascending the sides, take up its
residence immediately above the edge of the water with its shell wholly
exposed. It is a tranquil species, seldom moving much about, and
never walking on the under side of the surface of the water. Where
it occurs at all, it is generally in profusion.

Dr Leach appears to have raised this species to the rank of a distinct
genus, under the name of Huglesa, but it hardly shows sufficient
peculiarities to warrant this step. The shell is certainly somewhat in-

termediate in form between that of Cyclas and Pisidium.

Sp. 3. P. nitidum, Nobis.—Tas. xx. Fig. 7, 8.
P. testa orbiculato-ovali, nitidissima, tenuiter striata; umbonibus obtu-

siusculis, striis paucis profundioribus.
Long. 14 lin. Alt. vix 1} lin. Crass. 1 lin.

Animal album; siphone brevi, infundibuliformi, apertura patula, plus

minusve margine crenato, plicatulo,

Testa minimé variabilis, orbiculato-ovalis, partim inequilateralis; przece-
denti pauld convexior, et pro ratione longitudinis altior; albo-lutescens,
nitidissima, rarO aut nunquam sorde aut rubigine obtecta, tenuiter
striata, striis hie illic, pracipué 3—5 umbones transeuntibus, dis-
tinctitis incisis: umbones obtusiusculi, dorsalem marginem pauld supe-

rantes.

British species of Cyclas and Pisidium. 305

I can no where find any allusion to this species, which though
similar to the two last in the general form of the shell, may at once
be distinguished from both, if attention be paid to the animal. I have
examined upwards of an hundred specimens from different localities,
and in every instance it has preserved its characters. Its chief pecu-
liarity consists in the formation of the siphonal tube, which is regularly
funnel-shaped, with the aperture very patulous, somewhat plaited at
the margin, and more or less crenate——These appearances are not always
obvious, unless the siphon is protruded by the animal to its utmost
extent: the mouth of the tube, which is rendered very dilatable in
consequence of the plaits, then becomes fully expanded, and the irregu-
larity of its partially reflexed margin is rendered distinctly visible—
The shell also, which is subject to scarcely any variation, is remarkable
for its extremely glossy hue and cleanly appearance, rarely presenting
any of that foulness with which the last and following species are so
often incrusted, although found inhabiting the same ditches; from which
circumstance it would seem to follow, that in the case of those species,
this is due to something more than a mere extraneous deposit from
the surrounding soil. It may also be distinguished by a few peculiar
striae drawn with great regularity across the wmbones near the apex of
each valve, and cut rather more deeply than the rest. This character,
however, will not be seen without a close examination. It is most
visible when the animal is alive and the glossiness of the shell remains
unimpaired; but even then it is sometimes necessary that this should
be held to the light and turned in different directions, in order that
the eye may catch the appearance in question. It is, however, more
or less obvious in every specimen that I have seen.

This species is widely dispersed throughout Cambridgeshire, in-
habiting various situations, though seemingly partial to clear water. It
is however seldom found in any great plenty.—I have also met

Vol. IV. Part Il. Qa

306 Mr Jenyns’s Monograph on the

with it in the ditches about Battersea Fields, and in other parts of
Surrey.

** Testa distincte inequilaterali.

Sp. 4. P. pulchellum, Nobis.—Tas. xxt. Fig. 1—5.
P. testa obliqué ovali, ventricosa, profundits striaté; umbonibus ob-
tusiusculis, simplicibus.—Fig. 1.
Long. vix 2 lin. Alt. 14 lin. Crass. 14 lin.
Var. 3.—Fig. 2, 3.
Plerumque minor, testa tenuits striata; umbonibus subacutis.
Long. 12 lin. Alt. 14 lin. Crass. 1 lin.
Pera pulchella, Leach MSS. in Brit. Mus.
Cyclas fontinalis, Brown in Edinb. Journ. of Nat. and Geog. Scien. 1.
11. Pl. 1. f. 5—7—Alder in Trans. Nat. Hist. Soc. Newcast. 1. 41.
Var. y.
Testa obliqué ovali, tenuiter striata, compressa, marginibus acutis.
‘ Long. 14 ]in. Alt. 14 lin. Crass. ? lin.

Var. 3.—Fig. 4, 5.

Testa suboblonga, ventricosissima, profundius striata; margine obtu-
sissimo.
Long. 14 lin. Alt. 14 lin. Crass. 14 lin.

Animal album, siphone polymorpho; cylindraceo, conico apice truncato,
vel obconico; nune abbreviato, nunc in tubum gracilem subelongatum
(precipué in Var. 3.) extenso; margine hic illic inciso, vel inte-
gerrimo.

Testa quam maximé variabilis, in a, B, et y, obliqué ovalis, distincte
inzequilateralis, nune ventricosior, nunc compressiuscula, plus minusve

profundé striata, nitida, cinereo-lutescens, interdum autem sorde ferru-

British species of Cyclas and Pisidium. 307

ginea omnind incrustata: umbones simplices projectura nulla, plerum-
que obtusiusculi.
Var. 6. suboblonga margine dorsali subrecto, minus inequilateralis, ven-

tricosissima, margine basali obtusissimo.

This species was originally discovered by Professor Henslow, and
sent by him many years since to Dr Leach, who gave it the above
name. I have since met with it in plenty throughout Cambridgeshire,
and likewise in other parts of the county. In fact it is one of the
most common species *, inhabiting rivers, ditches, and likewise the smallest
streams: it is rather active in its habits, frequently ascending the sides
of the vessel in which it is confined, but I never observed it to walk
along the under side of the surface of the water. The siphonal tube
assumes a variety of appearances even in the same individual, and it
is very interesting to watch, under a low power of the microscope, the
striking and rapid changes of form through which it passes in a short
time. It is altogether a variable species, and the shell is of a very
different character in different situations; yet from the circumstance of
my possessing many intermediate specimens, I feel confident that the
above are only varieties—Var. 0. is from a pond on Bookham-Common
in Surrey; the others are all of frequent occurrence, and are often so
much incrusted over with a kind of ferruginous earth as to be entirely
concealed, and to present more the appearance of seeds or small lumps
of dirt than that of shells. Whether this is the effect of soil and
water, as seems to me more likely, or has any thing to do with the

habits of the species, is not very obvious.

* The discovery of this and some other minute species, which though of frequent
occurrence remained for a long time unnoticed by Conchologists, may be attributed to the
use of a peculiar net invented by Professor Henslow about the year 1815. This instru-
ment being constructed of the finest wire gauze enables the collector to strain the water
more thoroughly than by any other method previously attempted: and thereby to separate
the very smallest shells from the mud in which they are immersed.

QQ2

308 Mr Jenyns’s Monograph on the

There can scarcely be a doubt that this is the ménute shell found
by Montagu (Test. Brit. 88.) which he confounded with the young of
P. amnicum. Mr Alder sent me a small variety of it from Newcastle
as his Cyclas fontinalis, and I do not feel certain that it is not the
Pisidium fontinale of Pfeiffer whose characters in some respects aceord
better with this species than with P. pusillum already described.

My largest specimens of this shell are from the neighbourhood of
Battersea Fields.

Sp.5. P. Henslowianum, Nobis.—Tas. xx1. Fig. 6, 7.
P. testa obliqué ovali, ventricosa, tenuiter striaté; umbonibus sub-
acutis, projectura lamelliformi adornatis.
Long. 24 lin. Alt. 2 lin. Crass. 14 lin.
Pera Henslowiana, Leach (olim.)
Tellina Henslowiana, Shepp. in Linn. Trans. 14. 150.
Pera appendiculata, Leach MSS. in Brit. Mus.
Cyclas appendiculata, Turt. Man. 15. f- 6.

Animal album, tubo siphonali brevi, quoad formam paulo variabili; ple-
rumque subconico, apice truncato.

Testa obliqué ovalis, ventricosa, anticé planiuscula, distincte ineequilate-
ralis, tenuitér striata, nitidé lutescenti-alba, vel cornea, sepits partim
precipué ad apicem, sorde ferruginea obtecta: umbonies acutiusculi,
projectura parva lamelliformi adornatis.

Obs. In pullis projectura medio valvularum insidet; hinc gradatim

assurgit, acerescente testa.—(Vide Fig. 8, 9.)

The discovery of this very peculiar and well marked species is like-
wise due to Professor Henslow, who first found it in ditches com-
municating with the river Cam in the immediate neighbourhood of, and

also a few miles below Cambridge. Dr Leach named it after him;

British species of Cyclas and Pisidium. 309

but subsequently changed the name to that of Pera appendiculata,
reserving the above specific name for another and larger shell. This
last having however since proved to be a mere variety of the following,
I have restored the name of Henslowianum to the present species,
which has indeed already been described under that title by the Rev.
R. Sheppard in the Linnean Transactions.—This shell is so strikingly
distinguished by the curious eave-like projection upon the wmbones, that
it cannot be confounded with any other. In quite young specimens
this projection is, as it were, a small wing arising from the middle of the
valves, but as the growth of the shell proceeds, this last receiving its
increase principally at the basal margin, it appears to mount higher up,
until at length in adult individuals it occupies quite the summit of the
shell, where it appears like a small ridge or lamina rising up vertically
on either side of the hinge.—In other respects this species is very similar
to the last; nevertheless it is always larger—I have met with it in
two or three parts of Cambridgeshire, but it does not appear to be of

very general occurrence. Sheppard found it in Suffolk.
Sp. 6. P. amnicum, Nobis.—Tas. xx. Fig. 2.
P. testa ovali, ventricosé, profundits sulcato-striaté; umbonibus ob-
tusiusculis.
Long. 5} lin. Alt. 33 lin. Crass. 2# lin.
Tellina amnica, Muller, 2. 205.—Gmelin, 3242.—Linn. Trans. 8. 60.
Dillwyn, 1. 105.—Turt. Conch. Dict. 168.

rivalis, Maton, in Linn. Trans. 3. 44. t. 13. f. 37, 38.—Donovan.
t. 64. f. 2.

Cardium amnicum, Montagu, 86.

Cyclas palustris, Draparnaud, 131. ¢. 10. f. 15, 16.
Cyclas obliqua, Lamarck, 5. 559.—Nilsson, 99.

310 Mr Jenyns’s Monograph on the

Pisidium obliquum, Pfeiffer, 124. 4.5. f. 19, 20.
Cyclas amnica, Twurt. Conchyl. Brit. 250. t. 11. f: 15.—Fleming, 453.
Turt. Man. 15. f. 5.
Var. p.
Sulcis profundius exaratis.
Pera fluviatilis, Leach, MSS. in Brit. Mus.
Var. y.
Striis levis impressis.

Pera Henslowiana, Leach, MSS. in Brit. Mus.

Animal album, siphone pauld variabili; nunc abbreviato, subconico, apice
obliqué truncato; nune elongato, cylindraceo, apice plus minusve

recurvo.

Testa pauld variabilis, ovalis, vel obliqué trigona, distincté ineequilateralis,
ventricosa, anticé planiuscula, pulchré striata hic et illic sulcis pro-
fundioribus; cinerascenti-fusca, maculis et zona marginali lata palli-
dioribus, interdiim nitidé lutescentibus; raritis omnino fuscescens aut
lutescens; intus cxrulescens: umbones obtusi, sorde ferruginea ut in

precedentibus szepe incrustati.

This species, which was first published as British by Dr Maton 1. ec.
is at once distinguished from all the others in this genus by its very
superior size. It is not uncommon in rivers and gently running streams,
residing wholly at the bottom, and being partially buried in the mud,
but I have not often observed it in perfectly stagnant waters. Vars.
B and y were sent to Dr Leach by Professor Henslow from the
neighbourhood of Cambridge. The former of these gentlemen con-
sidered them as distinct species, and they accordingly stand in the
collection at the British Museum, under the above names; but I am

perfectly satisfied that they are mere varieties, differing in nothing

British species of Cyclas and Pisidium. 311

but the depth and number of the longitudinal furrows and striae, than
which no characters can be more variable. The shell of this species,
like that of the others belonging to this section, is frequently incrusted
over with a kind of ferruginous earth, which prevails chiefly at the
posterior extremity. Perhaps in the present instance this circumstance
is connected with the habits of the animal; which usually having the
anterior half of the shell fixed in the mud, the posterior and exposed
portion receives all those finer particles of the soil which drifting
downwards with the stream, are thereby deposited on its surface.—
The young of this species are readily distinguished from the two last,
by their more compressed shell, with the umbones scarcely at all promi-

nent and the striz more distinct.

The foregoing list includes all the British species belonging
to the above genera which I have been able to identify satis-
factorily. I possess one or two other shells which appear dif-
ferent from all hitherto described, but not having seen a sufficient
number of specimens to judge of their true characters, I should
not feel authorized in admitting them as really distinct.—I men-
tion the circumstance, however, for the sake of exciting further
enquiry upon the subject.

LEONARD JENYNS.

SwaFFHAM BULBECK,
Nov. 14, 1831.

Fic.

Fic.
Fie.

Fic.

Fic.
rc:
Fic.
Fic.
Fic.
Fic.
Fic.

Fic.

Fic.
Fic.
Fic.
Fic.
Fic.
Fic.
Fic.
Fie.

Re BON eae Sa

ORR Su SREY Sos tO

312

EXPLANATION OF THE PLATES.

PuLatTE XIX.

Cyclas calyculata ; Showing the general appearance of the shell and animal
inhabitant in that genus.

Pisidium amnicum. Ditto.

Hinge of Cyclas calyculata. A, the right, and B, the left valve; a, the
cardinal teeth,

Hinge of Pisidiwm amnicum. A, B, and a, as before.

PuLatE XX.
Pisidium obtusale.

Ditto ; as it appears when walking on the under side of the surface of the water.
Ditto; viewed from one end.

and 5. Pisidiwm pusillum. 'Two extreme varieties.

Ditto; viewed from one end.
Pisidium nitidum,
Ditto ; viewed from one end.

PLATE XXI.

Pisidiwm pulchellum: a, 6, and c, different appearances of the siphonal
tube.

Ditto, Var. 8: d, e, f, different appearances of the siphorial tube.
Ditto; viewed from one end.

Ditto, Var. 0.

Ditto ; viewed from one end.

Pisidium Henslowianum.

Ditto; viewed from one end.

Ditto, young.

Young; viewed from one end.

Oss. All the above figures are highly magnified.

aa . , 2 ly 4 ° y
hantackiord of the toes ‘ Gy, Pcs Lod b 4 iy

ns pre
Dig :

o : 2 : yiee
a ED, Vie She Caml : Gol, 3 Ps 7, ot. dc

P , ; , . [ RG of Me emt Fh. pt Vp. 4. YY yy

XI. On a new Analyzer, and its Use in Experiments
of Polarization.

By G. B. AIRY, M.A. F. R. Ast. Soc. F. G.S.

LATE FELLOW OF TRINITY COLLEGE, AND PLUMIAN PROFESSOR OF ASTRONOMY AND
EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE.

[Read March 5, 1832. ]

On two occasions I have had the good fortune to lay before
this Society anticipations of optical phenomena founded on theo-
retical considerations, which have been fully veritied by experi-
ment. This agreement I consider important, not because the
phznomena possess any intrinsic value, but because the precise
coincidence of observed appearances with theoretical calculations
affords the strongest possible proof of the correctness of the
theory. I have now to offer another instance of the same kind:
in which the experiment was suggested solely by theoretical con-
siderations, and in which the appearances, so far as I can observe,
agree perfectly with those which theory had indicated. Like the
others, it appears to me to give strong evidence of the correctness
of all the fundamental assumptions of Fresnel’s theory.

The experiment was first suggested by considerations of the
most general kind respecting the use of the analyzing plate in the
common polarizing apparatus. When polarized light (whether
plane, circular, or elliptical) has passed through a crystalline
plate, its intensity, by theory as well as by experiment, is the

Vol. IV. Part II. Rr

314 Proressor Arry on a new Analyzer,

same as before it entered. The use of the analyzing plate is to
resolve the emergent light according to some general law into
two parts, of which one is totally suppressed and the other (at
least a definite part of it) is wholly reflected.to the eye. As the
crystalline plate has at incidence resolved the incident light into
two parts which at emergence it has united with different degrees
of retardation according to the direction in which the light was
incident, the nature of the emergent light is different according to
the direction in which it was incident, or (which is the same)
according to the direction in which it emerges: and when resolved
according to the general law above mentioned, the proportion
of the reflected part varies according to that direction: and hence
the various intensities of the light in different parts of the image
seen with the ordinary apparatus. And this may be considered
as the general explanation of the use of an analyzer, including in
this term the common analyzing plate as well as such combinations
as will be hereafter described.

By the common analyzing plate (an unsilvered glass reflector
at the polarizing angle, or a plate of tourmaline, or a doubly re-
fracting prism considered with reference to one pencil only) the
emergent light is resolved (according to Fresnel’s theory) into two
sets of vibrations, one parallel to and the other perpendicular to
the plane of polarization of the analyzing plate;‘the former of
these is wholly suppressed, and the latter is wholly transmitted to
the eye. And this is the only kind of analyzation, so far as
I know, that experimenters or theorists have ever yet considered.
But there is not the least need for confining ourselves to this
method of analyzation. There are other methods, quite as simple
in theory, and nearly as easy in practice, which effect a resolution
of a very different kind. As the first among these I may mention

and its Use in Experiments of Polarization. 315

that it is easy to conceive and to treat theoretically a resolution
of the light emerging from any point of the crystal into two
pencils, one circularly-polarized and right-handed, and the other
also circularly-polarized but left-handed. To exhibit practically
the effects of this it is only necessary to contrive an analyzer
which shall wholly suppress right-handed circular light and wholly
transmit left-handed circular light, or vice versa. Another more
general resolution is into two elliptically-polarized pencils, one
right-handed and the other left-handed, the axes of the ellipses
having the same proportion, but the direction of the major axis
of one coinciding with that of the minor axis of the other. I am
not prepared to say whether or no any other resolution will be
found practicable.

Now conceive the analyzer to be of the first new kind that
I have mentioned, namely to have the power of wholly suppress-
ing right-handed cireular light and of wholly transmitting left-
handed circular light, or vice versa. I shall not consider the case
of plane-polarized light incident on the crystalline plate, because
the appearances are almost exactly the same (in theory as well as
in experiment) as when circularly polarized light is incident on
the crystal and the emergent light is analyzed by the common
analyzing plate: which case I have fully considered in former
memoirs in the Transactions of this Society, as well as in another
work. But let us consider the case of circularly polarized light
incident on the crystal, and after emergence analyzed by our new
analyzer. The first idea that strikes us in this combination is,
that there is nothing, except in the crystal, which has any respect
to sides. For the only incident light is circularly polarized: the
only light allowed to emerge is circularly polarized. The ap-
pearance therefore of the coloured rings, &c. must be such as

RR2

316 Proressor Airy on a new Analyzer,

conveys no trace of any plane of polarization: and must not vary
as the crystal or the analyzer is turned round.

In the common exhibition of the coloured rings (the incident
light being plane-polarized, the analyzer being the common ana-
lyzing plate, and the inclination of the planes of polarization
being any whatever) the principal trace of the planes of polari-
zation is in the uncoloured brushes. In uniaxal crystals they
form an eight-rayed star, composed of two square crosses inclined
at an angle equal to that between the planes of polarization,
every ray of which separates complementary rings. In_biaxal
crystals they compose two pairs of rectangular hyperbolas, the
angle between whose asymptotes is the same as that between the
planes of polarization, and whose branches divide complementary
rings. The two crosses or the two sets of hyperbolas unite when
the planes of polarization are parallel or perpendicular,

The first conclusion then is that, in the case under consider-
ation, the rings exhibited by crystals will not be traversed by any
brushes. Plates of Iceland spar, if they exhibit any variations of
light at all, will exhibit circular rings without a cross: and plates
of biaxal crystals will exhibit complete lemniscates without any
interruption from curved brushes.

The next conclusion is that the brightness of the light at
the poles of the image will depend only upon this consideration ;
whether the direction of the circularly polarized light which is
incident on the crystal is the same as the direction of that which
the analyzer can transmit to the eye, or is the contrary. If it is
the same, since the light which forms the poles of the image is
that which is not separated into an ordinary and extraordinary
ray, and therefore passes unaltered through the crystal, then the
light incident on the analyzer is exactly of the kind which the

and its Use in Experiments of Polarization. 317

analyzer can transmit to the eye, and therefore the pole is seen
with full brightness. If it is the contrary, the light incident on
the analyzer is exactly of that kind which is totally suppressed
at the analyzer, and therefore the pole is perfectly black.

The third conclusion is that (supposing, to fix our ideas, that
the direction of the light incident on the crystal is the same as
that which the analyzer can transmit) the intensity of light de-
pends only on the gain or loss of the ordinary or the extraordi-
nary ray: being at its maximum when that gain or loss is a whole
multiple of A, and nothing when the gain or loss is an odd

multiple of x. For the first of these propositions it is only

necessary to state that the crystalline plate, having resolved the
incident light into two waves consisting of vibrations of different
kinds, and having retarded one set more than the other by a whole
multiple of A, unites the two sets again in exactly the same cen-
dition in which they were at the resolution, and therefore they
emerge forming a kind of light which is exactly similar to the
incident light, and which is on that account susceptible of perfect
transmission by the analyzer. This applies to quartz as well as
to other crystals. For the second proposition we have only to
remark that when circularly polarized light is incident on Iceland
spar, nitre, and similar uniaxal and biaxal crystals, whatever be
the position of the planes of polarization, it is resolved into two
sets of plane vibrations at right angles to each other, one of which

is ; behind the other: and that when the relative path of these

9° that which preceded

waves is altered by an odd multiple of
by ; now follows by * (neglecting multiples of \): and the direction

of the circularly-polarized light is thus reversed: and the emergent

318 Proressor Arry on a new Analyzer,

light is therefore exactly of that kind which cannot be transmitted
by the analyzer, or the corresponding point of the rings is black.
This evidently will not apply strictly to quartz: but it will apply
with tolerable accuracy when the rays make a considerable angle
with the axis of the crystal.

The properties of Fresnel’s rhomb suggest at once a method of
constructing such an analyzer as we require. It is well known
that if circularly-polarized light is incident on Fresnel’s rhomb,
it emerges plane-polarized, and the position of the plane of po-
larization at emergence makes an angle of + 45° or — 45° with the
plane of reflection according as the incident light was right-
handed or left-handed. Let the light emerging from the rhomb
be received on an unsilvered glass at the polarizing angle, whose
plane of reflection makes the angle + 45° with that of the rhomb.
Now it is plain that if the light incident on the rhomb was
right-handed, it becomes plane-polarized in the plane of reflection
of the glass, and therefore is wholly reflected: if it was left-
handed, it becomes plane-polarized in the plane perpendicular to
the plane of reflection of the glass, and therefore is wholly sup-
pressed. This combination therefore (a rhomb and an unsilvered
glass at + 45°) has the property of wholly transmitting right-handed
circular light and of wholly suppressing left-handed circular light:
and in the same way it would appear that the combination of
a rhomb and an unsilvered glass at — 45° has the property of
wholly suppressing right-handed circular light and wholly trans-
mitting left-handed circular light. Since all polarized light (and
therefore light in general) may be represented by two pencils of
opposite circularly-polarized light, it follows that our combination
will have the power of resolving all light into two such pencils
and suppressing one of them.

and its Use in Experiments of Polarization. 319

In practice, the use of Fresnel’s rhomb in this part of the
apparatus would (on account of its length) be attended with some
inconvenience. I have therefore preferred for this purpose a plate
of mica, of such a thickness that the ray polarized in the plane

of one of its principal sections is retarded either 7 = or 5 of

a wave (according to the convenience of splitting) more than that
polarized in the plane of the other. The mica being attached to
the unsilvered glass so that its principal section makes an angle
of 45° with the plane of reflection, an analyzer is produced which
answers the same purposes, in general, as that described above.
In strictness its effects are not the same, as the order of the
colours is in some cases sensibly disturbed.

Upon trying this, when the incident light is circularly polarized,
the general effects are precisely such as were anticipated. Iceland.
spar exhibits rings without a cross of any kind: nitre, arragonite,
&e., exhibit the lemniscates uninterrupted in their whole extent,
and without any trace of hyperbolic brushes. Unannealed glass
exhibits dark patches surrounded by colours of different orders,
without any continuous brush. In these and in other cases that
I have tried, no alteration is produced in the appearances by turn-
ing the crystal, except that the system of rings, &e. is equally
turned.

Perhaps the method of analyzation which I have described
may, with the application of circularly-polarized light incident,
be advantageously used for examining the nature of irregularly
crystalline bodies. For instance: the appearances presented by
unannealed glass in the common apparatus are singularly com-
plicated: but in this they are comparatively simple. In this the
eye sees at one glance (by the order of colour) how much the

320 Proressor Airy on a new Analyzer,

ray polarized in one plane is more retarded than the ray polarized
in the plane at right angles to it. The position of these planes
however is not defined: that will be best found by observing
the dark brushes in the common apparatus.

To give a mathematical form to the investigation, let us dis-
tinguish the two kinds of circularly polarized light by the letters
A and B. Suppose that light of the kind 4 is incident, and that

its vibration is resolved into + a@.sin - (vt —.«) for the ordinary
ray, and + @. cos an (vt—.2) for the extraordinary. On emerging
from the crystal they may be represented by + a.sin == (vt — 2)
for the ordinary ray, and + a. cos = (vt— «+ 0) for the extraordi-
nary. These will be separated by the analyzer into two pencils

of light of the kinds 4 and B; these will be represented by

_ Qa : incl
+ p .sin (vt — x + q) perpendicular to the principal plane constituting

+ p cos == (vt — x + q) parallel to the principal plane light 4,
and

, . 6) Tv is : i. ’
+ p’.sin ce (vt — x + q’) perpendicular to the principal plane | donstteating

— p'. cos a (vt — x + q) parallel to the principal plane | light B.

Making the vibrations in each plane equal to those at emergence
from the crystal,

and its Use in Eaperiments of Polarization. $21

Or since this must be true for all values of v¢ — 2, we may expand
the sines and cosines and equate the coeflicients of

5 ees 2
sin oo (vt — x) and cos = (vt — 2).

Thus we have ' é
a= p.cosg+ p'cosg

O= p.sng+p'sing

—asnO=—p.sing +p’ sind

acsO= p.cosq—p'cos”.

From these,

iS) ; :
p=ac0s>5, p=asin

and the vibrations are

SN oe ice ts) :
@ cos 5 .sIn > — @ — «#£+ 5) perp. to prine. pine
constituting light 4,
see tas we re) even Wee |
@ COS 5. COS > (© = a 5) par. to prine. plane
Nes ON... Big: 0)
asin >.sin — (vt — x + 270° + —) perp. to prine. plan
2 r ( Gisliaca 2) et papiicupae Gs I constituting
ke 2 . light B.
— @ Sin 5 . cos = (ve — x + 270° + 3) par. to prince. plane | 7

If the analyzer is of such a kind that it can transmit light 4,

the intensity of the light that reaches the eye is 2@ cos? 5 : if

it transmits light B, the intensity of light is 2 a’ sin’ 5 . The former

of these gives light at the place where there is no double re-

fraction: the latter gives dark at the same place. But in both

cases it is plain that the brightness depends only on 0, the

quantity which one ray has gained or lost on the other, and that

it does not at all depend on the position of the planes of polari-
Vol. IV. Part II. Ss

322 Proressor Airy on a new Analyzer, §e.

zation: and therefore in both cases the appearance will be that
of patches or curves extending continuously through all the parts
where the gain of one ray upon the other is a constant quantity.

In the former part of this paper I have alluded to an analyzer
of amore general kind, namely one in which the light is separated
into two elliptically-polarized rays. This is constructed by plac-
ing the plate of mica with its principal plane inclined to that
of reflection at the unsilvered glass by an angle different from
45°. The investigation of its effects is not more difficult than that
above, but is rather longer, and its results will hardly justify its in-
sertion. I will only remark that, with the apparatus which I have
supposed employed for the experiment above, if the Fresnel’s
rhomb or mica by which the incident light is made circularly-
polarized, and the mica by which the new analyzation is effected,
are turned the same way (leaving the glass reflector unmoved)
the continuity of the rings is not interrupted, but a part of the
image is seen to grow darker and darker; and when both are
turned 45°, this dark part becomes the black brush. This sup-
poses the planes of original polarization and of reflection at the
glass to be at right angles; but if they are parallel the change
is of the opposite kind, and the bright brushes are finally pro-
duced. ‘

G. B. AIRY.
OBSERVATORY,
Jan. 19, 1832.

EE,

—_——

TRANSACTIONS

OF THE...

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

Vou. [V. Parr ILI.

XII. On the Mechanism of the Larynx.

By ROBERT WILLIS, M.A. F.R.S. F.G.S.

FELLOW OF CAIUS COLLEGE, AND OF THE PHILOSOPHICAL SOCIETY.

[Read May 18, 1829.]

IT must be a source of great regret to those who have made
themselves acquainted with the present state of Acoustics, to find
the various investigations concerning the mechanism of the hu-
man voice leading to such unsatisfactory and even contradictory
results. For from whence may more instructive lessons in that
science be expected than from an apparatus which is capable of
producing sounds in every variety of pitch, quality, and intensity,
from the most exquisite music to the most execrable noise; an
apparatus of no extraordinary dimensions, and one moreover of
which the greater part is exposed to our observation during the
various changes of ‘form which it assumes whilst in action. Never-
theless the laws which connect these changes of form with the
production and yariation of the sounds are hitherto obscure. To
account for this, we are compelled to refer to two considerations ;
on the one hand, the instrument of the voice is not exclusively
appropriated to its production, but is also evidently adapted to the
performance of functions far different and more important to the
animal economy; and, on the other, the explanation of the phe-

Vol. IV. Part III. SAG,

324 Mr WILtIs on the

nomena, in as much as they are produced by a part of the animal
frame, has been consigned, with that of its other functions, to the
Anatomist and Physiologist, to those whose professional studies
are completely unconnected with Acoustics, a science which in
all investigations of this kind must necessarily take the greatest
share.

Accordingly every treatise on Physiology or Anatomy contains
a chapter on the organs of voice, in which the parts conducing to
its formation are described; whilst the contradictory and some-
times careless accounts which the best anatomical writers give of
the mechanical action of these parts, and of the mode in which
they perform their functions, form a vexatious contrast with the
minute accuracy of their anatomical descriptions.

In the present memoir I have attempted a more minute analysis
of a part of these organs than appears to have been hitherto un-
dertaken. As, however, I am writing for philosophical readers
in general, I have purposely divested my descriptions of the tech-
nical form as much as possible, and my drawings are to be regarded
more as plans or types of the general structure gathered from the
examination and comparison of many, than as representations of
any one individual.

The vocal mechanism may be considered as |

consisting of Lungs or Bellows, capable of trans- aaa!
mitting by means of the connecting Windpipe a ey
current of air through an apparatus contained in the :
upper part of the Windpipe, which is termed the 3
Larynx. This apparatus is capable of producing es
various musical notes which are heard after passing vey i.
through a variable cavity, consisting of the pharynx, Bellows.

mouth, and nose.

Mechanism of the Laryne. 325

Now, if this arrangement be artificially imitated by combining
together pipes and cavities with bellows in a similar order, and
substituting for the Larynx any elastic lamina capable of pro-
ducing musical notes when vibrated by the stream of air, it is
found that by changing the form of the cavity above it, the various
qualities which distinguish the continued notes of the human voice
im speech, may be so nearly imparted to the sound which the
imitative Larynx is producing, as plainly to shew that there is no
necessity for seeking any power of altering the quality of the notes
in the Larynx itself. This then may be considered as merely an
instrument for producing certain musical notes, which are after-
wards to be converted into vowels, liquids, &c. by the proper
changes of form in the superior cavity.

We may here remark an essential difference between the vocal
mechanism and our ordinary musical wind instruments, which are
generally made up of some vibratory mouth-piece to generate the
note, and an attached cavity, or pipe, to govern and augment its
tone, each instrument having its peculiar quality ; whereas the
attached cavity in the vocal machine is capable not only of govern-
ing and improving the musical quality of the note, but also of
imparting to it all manner of various qualities, the numerous
vowels and liquids of speech, and also the perfect mimicry of the
peculiar sounds of nearly all animals and musical instruments.

Of this cavity it is not my present purpose to speak. In-
deed I doubt whether the science of Acoustics is sufficiently ad-
vanced to enable us completely to understand its mechanism. IT
shall in this memoir confine myself to the Larynx.

The precise form of the Laryngeal cavity, and the parts im-
mediately connected with it, is shewn in Figs. 1 and 2, Fig. 2
is a section made by a plane passing through the nose, mouth,

ee 4

326 7 Mr WIt.ts on the

and windpipe, called the mesial plane, dividing the head sym»
metrically, DEFG being the external outline of the throat,
and H the back of the tongue. Fig. 1 is a section made by
a plane perpendicular to the former, and passing along the line
A: BC.

From A to D (Fig. 2) the windpipe presents a_ horizontal
section nearly circular; above D it contracts in the transverse di-
mension, assuming the form of a narrow slit, termed the glottis.
(The line GG (Fig. 1.) passes through the glottis). Immediately
above it the windpipe expands into a pair of cavities, termed the
ventricles of the Larynx, through which the line VV passes, and
above these the. passage again narrows into another slit indicated
by the line ZZ, which has been termed the pseudo glottis. Above
this the passage again expands, and finally opens into the pharynx,
as the cavity behind the tongue is termed.

The whole surface of the cavity we have been describing is
lined with a soft mucous membrane, similar to that which is seen
on the inside of the mouth, soft palate, &c., with the exception of
the edges of the glottis, where the lining assumes the form of a
ligament, white, fibrous, and elastic, the outline of which is seen
in Fig. 2, immediately below the opening of the ventricle. The
edges of the pseudo glottis are formed merely by a kind of redu-
plication of the ordinary mucous membrane. It will be seen from
Fig. 2, that neither the ligaments nor the ventricles extend entirely
across the passage.

The most generally received opinion, and that which appears
to me to be borne out by a careful investigation of the structure
of the Larynx, is that the current of air from the lungs excites
these ligaments to vibration, and so produces the sounds of the
voice (vide note A). Hence they are denominated the vocal liga-

Mechanism of the Larynx. 327

ments. I shall now proceed to a more minute examination of the
precise nature of this vibration, and of the mechanism of the Larynx
generally than appears to have been hitherto attempted. Assum-
ing then that the source of the notes of the voice is to be found
in the vibrations of a pair of membranous elastic edges, between
which a current of air is allowed to pass, I shall endeavour to
shew under what conditions such elastic edges must be presented
to a current of air, in order that it may elicit from them the re-
quired vibrations.

One of the most commodious ways of investigating this is to
prepare a piece of wood of the form ABCD (Fig. 9) and _ paste
on one side of it a piece of fine kid leather, the upper end of
which (mn) is cut straight, and a moderate degree of tension given
to the leather when pasted on.

This arrangement presents us with an elastic membrane, whose
upper edge mn is free and the other edges confined, and, therefore,
with a case analogous to that of the vocal ligaments.

EF (Fig. 10) is the plan of a flat board, having a rectangular
opening GH in the middle, and LM (Fig. 12) is a vertical section
of this board along the line IK, (Fig. 10,) shewing a pipe N
attached to the lower side of the board, in order to connect it
with a pair of organ bellows, by means of which a current of
air may be maintained through the rectangular opening at pleasure.
(In Figs. 14, 16, and 18 are similar sections).

Let now an upright board OP, (Fig. 12,) be clamped upon
LM, so that its face P may coincide with the side of the opening
throughout its whole length. If the leather in its frame, (Fig. 9,)
be exposed to the action of the current by placing the lower edge
BC of its frame in contact with this board and always parallel
to the side of the opening, and if then the frame be turned on

328 Mr Wits on the

its lower edge, so as to place the plane of the leather at different
angles with that of the board, the following phenomena will be
observed :

If the leather be inclined to the board, as in Fig. 12, the current
will merely drive it outwards and fix it with its upper edge con-
cave to the board, as in Fig. 11, (which is a bird’s-eye view of the
upper edge).

If the leather be inclined from the board, as in Fig. 14, the
current will draw the upper edge inwards, maintaining it with
its upper edge convex to the board, as in Fig. 13.

If, however, the leather be placed in the intermediate position
to these two, that is, parallel or nearly so to the board, the current
will excite and maintain strong vibrations in the upper edge of
the leather, producing a loud musical note, as long as the current
is kept up.

If for the board we substitute a similar frame with leather,
and apply the two frames opposite to each other, above the rect-
angular opening of the board, as in Figs. 16 and 18, we have an
arrangement somewhat resembling the glottis, in possessing a pair
of edges opposite to each other; with this similar phenomena to
those just described are observable; namely, when the leathers
are inclined fo each other, as in Fig. 16, the current maintains their
upper edges in the position Fig. 15. When they are inclined from
each other, as in Fig. 18, the current fixes them in the position
Fig. 17, but when nearly parallel puts them into strong vibration.
The angle at which they may be inclined to each other to make
vibrations possible varies with the tension of the leather and the
force of the current. For an examination of the reasons of these
phenomena I must refer to Note B; it is sufficient for our present
purpose to know that it is not merely necessary for the vibration

Mechanism of the Larynx. 329

of a pair of ligaments, like those constituting the glottis, that a
current of air be passed between them, but that their opposite
surfaces must also be placed in a given position with respect to
each other.

For instance, Fig. 5 is the ordinary position of the ligaments
GG, in which the breath passing between them could never pro-
duce a sound from them, they being inclined from each other.
Whereas, in Fig. 1, where they are parallel, the breath would
instantly excite vibrations in them.

Here we have a solution of a difficulty which never seems to
have occurred to former writers; that is, why the ligaments are
silent while the ordinary breath passes between them. It cannot
be because their tension is not sufficient, because I shall shew
that they are always in a state of tension nearly corresponding
to the pitch of the speaking voice.

To shew how the same pair of ligaments may produce various
notes, let a wooden pipe be prepared of the form Fig. 19, having
a foot C like that of an organ pipe and an upper opening, long
and narrow as at B, with a point A rising at one end of it. If
a piece of leather, (or, still better, of Hancock’s sheet India rubber,)
be doubled round this point and secured by being bound round
the pipe at D with strong thread, as in Fig. 20, it will give us
an artificial glottis with its upper edges GH, which will vibrate
or not, at pleasure by inclining the planes of the edges, according
to the previous experiments. A couple of pieces of cork EF
may be glued to the corners to make them mere manageable.
From this machine various notes may be obtained by stretching
the edges in the direction of their length GH; the notes rising
in pitch with the increased tension although the length of the
vibrating edge is increased. It is true that a scale of notes of

330 Mr Wits on the

equal extent to that of the human voice cannot be obtained from
edges of leather, but this scale is much greater in India rubber than
in leather, and the elasticity of them both is so greatly inferior to
that of the vocal ligaments, that we may readily infer that the
great scale of the latter is due to its greater elastic powers. To
obtain various notes from the glottis, therefore, it is only necessary
to vary its longitudinal tension after its ligaments have been placed
in the proper position.

As, however, during breathing the air passes freely in and out
of the lungs through the identical apparatus by which the notes
are produced, the passage we have been considering, or glottis,
must be capable of assuming the form of a large and free aper-
ture; since it is certain, from the freedom with which the air is
inhaled and exhaled, that it is not compelled to pass through so
small a slit as the glottis appears to be during vocalisation.

The passage is also capable of being shut so close by its own
small muscles that all the exertions of the powerful abdominal
muscles acting upon the diaphragm to compress the lungs and
condense the air in the trachea are not capable of forcing it
open.

The production of a musical note takes place instantaneously,
at the pleasure of the individual. The breath has been previously
traversing the passage in silence, and at our will some change is
immediately made in the larynx, which produces the note, and
this certainly depends upon something more than the mere closing
of the passage, because we can make the aperture of the passage
pass through all degrees of contraction up to absolute closing
during the expiration of the breath without producing any sound,
except the usual rushing noise of a forcible current of air passing
through a narrow aperture.

Mechanism of the Larynx. 331

The law of vibration which I have above explained renders
this more intelligible; for since it appears that unless the mem-
branous edges of the passage are placed nearly parallel they can-
not be made to vibrate, we have only to suppose that the change
we feel in the Larynx is the placing of the ligaments in a parallel
position, and the whole mystery is explained. If therefore I can
succeed in shewing that the arrangement of the cartilages and
muscles is adapted for the purpose of placing the vocal ligaments
under the varicus conditions which have been shewn to be neces-
sary, I shall have done all that is possible to complete the
evidence in favour of my explanation.

For, after all, no explanation of the functions of a machine,
of which essential parts are concealed while in action, can be com-
plete and uncontrovertible: when we have examined the separate
parts, and have enumerated the functions which observation shews
the machine to be capable of performing, and by comparing these
with the different portions here elicited, as we flatter ourselves, a
complete allotment of each function to its appropriate part of the
structure, we have only been in fact describing a machine of our
own contrivance, copied in form, and capable perhaps of per-
forming the same functions, but not necessarily identical with
the original, because we cannot certainly know whether it per-
forms the same motions for the same functions. Hence we can
ouly establish a probability that the uses of the corresponding
parts in the two are the same. Beyond this probability we can
never get, unless we can succeed in viewing the machine in
motion.

This is not entirely the case with the Larynx, because we are
enabled to trace the motions of some of the cartilages from without ;

but the greater part of the machine is, and always must be, hidden
Vol. IV. Part III. Uu

332 Mr Wits on the

from our view while living, for we cannot make much use of the
facts said to have been observed by some Physiologists in their
experiments on living animals, which are reported by men plainly
but loosely acquainted with Acoustics, and which, as they have
been deduced from a vocal mechanism vastly inferior to the hu-
man, may very probably mislead us if we attempt to apply them
to the explanation of the latter.

Having now, in some degree, considered the uses of the Larynx,
and laid down some principles, we may proceed to examine its
structure with more minuteness.

Upon removing the mucous membrane which lines the whole
of the interior of the laryngeal cavity, but leaving the vocal liga-
ments in their place, we find the latter supported in a curious
frame of cartilages united by certain ligaments and articulations,
and provided with muscles, by the action of which the cartilages
may be made to assume various positions with respect to each
other, and thereby alter the tension and relative position of the
vocal ligaments.

The windpipe is found to consist of a pile of cartilaginous
rings, serving to keep the passage from the lungs always open, and
forming in this respect a contrast with the cesophagus, or tube
leading from the cavity of the pharynx to the stomach, which is
always closed by its muscular contractile structure, excepting at
the moment of the passage of food. In investigations concerning
the organs of voice the esophagus may always be regarded as
having no existence. Its place is indicated in Fig. 2 by the
line IK.

Above the rings of the windpipe, however, is a stout bony
annulus, denominated the cricoid cartilage, which serves as the
foundation of the mechanism we are about to describe.

Mechanism of the Larynx. 333

Fig. 3 isa section of the Larynx similar to Fig. 2, but repre-
senting it as stripped of its mucous membrane, &c., leaving the
bare cartilages with the vocal ligament in its proper place, and
also the muscles.

Fig. 4 is an external view of the corresponding half of the
Larynx, and Fig. 7 a bird’s-eye view of the entire Larynx, both
in the same state of dissection. Fig. 8 is an enlarged sketch of
part of the upper half of Fig. 7. In these four figures the same
parts are indicated by the same letters.

The cricoid cartilage is seen (Figs. 8, 4, ABC,) surmounting
the rings of the windpipe. The thyroid cartilage ECGH embraces
the cricoid, and is articulated to its sides by its lower horns at
C, so that it may be regarded as turning round the point C as
a fulcrum. As this discussion merely regards the motion of the
cartilages among themselves, it is of no consequence whether we
regard the cricoid or thyroid as fixed, and for convenience I shall
assume the cricoid as fixed for the present.

Upon the upper surface of the back part of the cricoid are
seated two small cartilages (°F, Figs. 7, 8,) termed the arytenoids.
They are placed upon articulating surfaces which are formed on
the upper outer edge of the cricoid, and which may be considered
as portions of cylinders, whose axes are inclined, both with respect
to the horizontal and vertical sections. In the vertical section
the projection of this articulating axis is in the position BL, Fig. 3,
and in the horizontal in the line OP, Fig. 8. The base of each
arytenoid is spread out, and curved below Q, Fig. 8, so as to
lie upon this articulating surface, to which however it is so loosely
adapted as to permit a small degree of sliding motion transverse
to the axis. The arytenoids are however firmly tied to the back
of the cricoid by a bundle of strong ligaments BR BS, Fig. 8,

uU2

334 Mr WItts on the

and BF, Fig. 3, diverging from the pomt B, Fig. 8, which point
B is as nearly as possible the point where the axis of the articu-
lating surface would intersect the cricoid. The vocal ligament
is stretched from the thyroid at J to the arytenoid at V, and as
there is no muscle which can relax the ligament BRS*, it re-
ceives and transmits to the point B of the cricoid the tension of
the vocal ligaments.

The motion of the arytenoid is therefore compounded of a rota-
tion round the axis OP, and of a slidmg motion transverse to
this axis, which is confined by the tension of BRS to a swinging
round the point B, of which we shall presently see the use.

We have already seen that the thyroid is so united to the
cricoid as to turn round the point C, Figs. 3 and 4, as a fulcrum.
The effect of this rotation is to alter the distance between the point
E of the thyroid, and B of the cricoid, and therefore to affect
the tension of the vocal ligaments.

If this distance be increased by the thyroid revolving in the
direction from B to E, the tension of the vocal ligament is in-
creased, and by its pulling at the arytenoid cartilage the tension
of the bundle of ligaments is increased. If the distance EB be
diminished by the thyroid turning in the reverse direction, the
contrary effect will take place.

To produce this motion, two pair of muscles are provided,
one of the external pair (the cricothyroid muscle) is seen at AK,
Fig. 4+, when this muscle contracts it brings the point K of the

* Vide Note C.

+ Each muscle of this pair is sometimes seen divided into two, and is described by
some writers as such. Some of the fibres are attached so close to the fulcrum C as to be
apparently intended to stretch the ligaments which bind the horns C of the thyroid to the
cricoid, and thereby unite more firmly these two cartilages during vocalisation.

Mechanism of the Larynx. 335

thyroid nearer to the point A of the cricoid, and therefore in-
creases the distance EB; this pair of muscles therefore stretches
the vocal ligaments.

One of the internal pair (the thyroarytenoid muscle) is seen
at Emae, Fig. 3; it is attached to the inside of the front of the
thyroid at Em, and to the arytenoid at ae; when this muscle con-
tracts it approximates the arytenoid to the pomt E, and as the
arytenoid is tied to the cricoid by the bundle of ligaments at B,
it of course draws the point B after it, just as if the muscle were
attached immediately to B. The effects of this muscle is then
to decrease the distance EB, and therefore to relax the vocal
ligament.

Hence the thyroarytenoid muscle is the antagonist muscle of
the cricothyroid, and together they govern the pitch of the notes.

The truth of this account of the stretching and relaxing of the
vocal ligaments may easily be verified, as far as the motion of
the cartilages is concerned, by a method which was first suggested
by Ferrein*, but appears to have been forgotten or misunderstood
by succeeding writers. We may readily trace with the finger on
the outside of the throat, (at GFED, Fig. 2,) the thyroid cartilage
EF, the cricoid cartilage DM, and a small space ED between them
(marked mn in Figs. 3 and 4). Now it is plain that when the
thyroid revolves upon C in the direction BE, so as to stretch the
vocal ligaments and raise the pitch of the notes produced, that
this motion approximates the lower edge m of the thyroid to the
upper edge n of the cricoid, and, therefore, diminishes the aper-
ture mn, and vice versa, when the ligaments are relaxed, the aperture

myn is increased.

* Ac. Par. 1741.

336 Mr Wits on the

But upon singing a scale of notes two motions are to be ob-
served in these cartilages; one is a general motion upwards, when
the pitch of the notes rises, and downwards, when it falls; which
we have no concern with at present, as we are treating only of
the motions of the laryngeal cartilages with respect to each other,
which this dees not affect. The other motion consists of the relative
motion of the cricoid JD and the point E of the thyroid, and con-
sequent variation in the distance DE, which is best to be traced
by lodging the tip of the finger in the little hollow between the
cartilages, and so following the general motion up and down.
By doing this carefully, the size of this aperture will be perceived
to follow a law exactly coinciding with the above explanation,
namely, always increasing with a descending pitch and diminishing
with a rising one.*

So far, the arytenoids have merely served as links, connecting the
vocal ligaments and thyroarytenoid muscles with the cricoid cartilage,
and the function just described would be just as well performed if
the vocal ligaments and thyroarytenoid muscles were attached to the
ericoid at B without their intervention. In fact, were the vocal liga-
ments merely intended to sound whenever the current of air passed
through the larynx these cartilages would apparently have no office.

But it is to be remembered, that to enable the ligaments to
vibrate they must be made to assume a peculiar position with
respect to each other, and that for breathing, it is necessary that we
have the means of opening the passage wide, also of entirely closing
it; during which it is essential that that peculiar position be avoided,
for fear of a sound being produced when not intended. It is in the
performance of all these motions that the arytenoids are concerned.

* Vide Note D.

Mechanism of the Larynx. 337

The articulation of the arytenoids with the cricoid has been
already described. From the extremity N, (Fig. 8,) of the ary-
tenoid arises a muscle, termed the cricoarytenoideus posticus, which
is turned round the edge of the cricoid, and affixed to the lower
part of the back of the latter cartilage. Its mechanical action,
however, is the same as if it acted on the arm of a short lever N,
in the direction NW on the plan, that is to say, perpendicular to
the axis of motion OP, and its effect is to produce rotation about
this axis, and therefore to separate the arytenoid cartilages from
each other and open the passage. (Vide Note C.)

From the arytenoid another muscle NX arises, and is attached
to the cricoid at and about the point X; this is termed ericoary-
tenotdeus lateralis. In Fig. 3, the fibres of this muscle may be
seen arising from X and passing up to the arytenoid, lying nearly
parallel to the projection of the axis of motion.

To understand the action of this muscle upon the arytenoid,
we must remember that the latter is attached to the point B by
ligaments, which radiate from this point, and are united to the
arytencid along its posterior surface from S to R.

The tension of this muscle then in the direction NX will, by
drawing the cartilage in that direction, stretch the ligament RB,
and tend to bring the points X NB into the same straight line; this
will at the same time approximate the point V to the medial plane
and corresponding point of the other arytenoid, and also, (as it
appears from Fig. 3 that m the vertical projection, N is above
the line joining BX,) it will depress N and still more V, because
the cartilage turns on the articulating surface beneath Q.

The effect, in short, of the pair of muscles in question is to press
the points V of the arytenoids together, at the same time depressing

them.

338 Mr Wits on the

The two arytenoids are moreover united by a muscle, called
the ¢ransversus arytenoideus, which arises from Rd of one ary-
tenoid, and is united to the other in the corresponding points. Its
section is seen at s in Figs. 2 and 8. It is removed in all the other
figures. Its action upon the arytenoids is plainly to press together
the point S and its corresponding one. Hence, when this muscle
acts at the same moment with the cricoarytenoidei laterales, which
we have been just considering, their joint effect will press the whole
of the adjacent faces of the arytenoids together, depressing the
points V and closing the glottis, and, therefore, antagonizing the
action of the cricoarytenoideus posticus. (Vide Note E.)

Indeed, it appears at once from the diagram that the forces NX
NY of these two muscles must together produce a resultant in the
direction nearly of WN, and therefore directly opposed to the action
of the cricoarytenoideus, which is represented in direction by NW.

Hence, the cricoarytenoidei postici open the glottis. The crico-
arytenoidei laterales and the arytenoideus transversus acting jointly
close the glottis.

The complete closing of that portion of the aperture which is
included between J’ and V, (Fig. 8,) appears to be effected jointly
by the motion of the arytenoid cartilages, which in closing together
approach the point 7, from the obliquity of their axis of motion,
and by the swelling of the muscle NX in contracting to bring the
arytenoids in contact ; both causes tending to compress the cellular
tissue and muscular fibres which occupy the space TX NV, and
therefore to close tightly together the sides of the passage below
the vocal ligaments.

We have now to consider the means by which the vocal ligaments
are placed in the proper relative position for vibration. ‘To explain
this, let (Fig. 6,) the continued line be the ordinary position of the

Mechanism of the Larynx. 339

glottis for breathing, when it is slightly opened. In this position
the vocal ligaments ab cd diverge from each other in such a manner
that, according to our previous experiments, the current of the
breath could never excite them to vibration, whatever their longitu-
dinal tension might be. If the points ac be carried upwards, at
the same time approaching each other, so as to acquire the position
ac, it is manifest that this change, by increasing the distances ea,
fe, will, by diminishing the convexities eba fdc, draw the passage
into the form indicated by the dotted lines eba’ fde’, in which the
vocal ligaments have assumed the position proper for vibration.
But the motion of the arytenoid round the axis OP, (Fig. 8,) which
we have already described, will, in raising the vocal ligaments,
separate them and take them rather into the position indicated
by the dotted line ea’.

It only remains then to explain how the extremity V, (Fig 8,) of
the arytenoid may be made to rise and approach the corresponding
point of the other arytenoid at the same time; for, if this is done,
the vocal ligaments will necessarily assume the required position.

This motion is permitted by the sliding of the articulating surface
of the arytenoid upon the cricoid, already described, and is effected
by the thyroarytenoidei muscles. These muscles we have shewn
to be only employed during vocalisation, and we shall now see
that their peculiar structure places at the same time the arytenoids
in the preper position for vibration.

The internal face of one of these muscles is seen in Fig. 3;
a bird’s-eye view of the opposite one is shewn in the lower haif
of Fig. 7 at kV. f. That corresponding to Fig. 3 is removed from
the upper half of Fig. 7, to shew the cricoarytenoideus lateralis
NX more distinctly, for a similar reason the latter muscle is re-
moved from the lower half to display the thyroarytenoideus hk Vf.

Vol. lV. Part IIT. X x

340 Mr Wits on the

In Fig. 3 the arytenoid and its attached vocal ligament and
muscle are in the vibrating position. When the arytenoid is in
the position corresponding to Fig. 5, the pomt F' is considerably
below the line EB. Hence when the thyroarytenoid muscle
is brought into action, its fibres, which lie on this face parallel
to the vocal ligament, tend of course to bring the points EF'B
into a straight line and hence raise the point F. This, by in-
ducing a rotation round the axis OP, would separate the liga-
ments were it not counteracted by the direction of the fibres of
the lower portion of the muscle, which arising from about m, near
the median plane are attached to the arytenoid at a much greater
distance from it; this may be seen clearly in Fig. 7. They,
therefore, draw the point N (Fig. 7, 8) of the arytenoid towards
the median plane producing the sliding motion so often alluded
to, whilst at the same time the upper fibres of the muscle, which
are not parallel to the lower, maintain the upper part of the carti-
lages in their due position with their points separated, so as to
part the upper ligaments of the glottis and keep them out of the
way of the current of air.

This may be elucidated by considering the muscle when in the
position corresponding to Fig. 5 as a very loosely twisted rope,
which when brought into action tends by untwisting itself to bring
its fibres into parallelism, and therefore to communicate a rotatory
motion to the attached arytenoid, which, combined with its articu-
lation to the cricoid, places it in the exact position required for the
vibration of its vocal ligament, which then assumes the form Fig. 1.

As the arytenoids are hidden from our sight, and cannot be
traced externally as the other cartilages can, it is plain that the
whole account I have given of their motions must be considered
as depending entirely upon induction.

Mechanism of the Larynx. 341

With respect to the scale of notes in the human voice, which is
termed the falsetto, I shall merely observe that it is at present ex-
tremely doubtful whether it owes its peculiar quality to some change
in the laryngeal mechanism, or in the superior cavity ; the motion
of the cartilages observed by the finger from without shews that
the tension and consequent diminution of the aperture ED, Fig. 2,
goes on in the production of these notes just as it does in that of
the natural tones, and is therefore carried so far in the higher notes
of the falsetto that the space ED is completely obliterated by the
upper edge of the cricoid touching the lower border of the thyroid.

According to M. Magendie* the vocal ligaments of a dog
vibrate through their whole length while producing deep notes,
but in high notes the hinder portion only vibrates, the thyroidean
extremities being closed together so as to shorten the aperture of
the glottis, this diminution of the glottis becoming greater and
greater as the notes rise in pitch. Should this ever be established
to be the case in man, I should not be surprised if it were found
that only during the production of the natural scale the vocal liga-
ments vibrate through their whole length, after the manner I have
described already; while for the production of the falsetto notes
the following changes may be introduced. If the arytenoids be
pressed together by the arytenoidei transversi and cricoarytenoidei
laterales, and at the same time lifted up into the vocal position
by the thyreoarytenoideit, the complete closing of the passage
will be prevented, and notes will be produced by the action of

® Ta poRi bs

+ The cricoarytenoidei laterales press the points V’ together (Fig. 8), at the same time
depressing them; but the thyroarytenoidei approximate the points V’, at the same time raising
them, and without bringing them into contact. If both these muscles act at once the raising
effect of the latter is greater than the depressing effect of the former, because the latter acts

x x2

342 Mr WI. Is on the

the current which will differ entirely in their quality from the
former, because the vibrating length of the ligaments will be di-
minished by the contact of the arytenoids, and they will beat
against each other during vibration.

The approach of the cricoid D to the thyroid at E, Fig. 2, will
however compress the cellular tissue, &c., and tend to press toge-
ther the vocal ligaments and the sides of the passage below them,
beginning at the thyroidean extremity of the glottis, and diminish-
ing the vibrating portion at that extremity ; and if the tension of
the ligaments be increased this compression will finally close
the aperture in the manner described by M. Magendie.

I have been more minute in examining the mechanical action
of the muscles than may at first sight have seemed necessary,
because for want of some such examination the greatest confusion
prevails in all the accounts of them. Thus, while all writers agree
that the cricothyroidei serve to approximate the cricoid cartilage
to the thyroid, either by raising the cricoid or depressing the thyroid,
none of them have shewn how these cartilages are to be separated
again, neither do they agree as to the effect of this approxima-
tion upon the glottis. Again, Cowper and Albinus make the thyro-
arytenoidei draw the arytenoids nearer together; but Sémmerring
and Haller make them separate these cartilages; and’ Meckel and
others make them draw forward the arytenoids. Haller thinks
that they relax the vocal ligaments; Bichat that they stretch them.
The cricoarytenoidei laterales are stated by Cowper, Haller, and

on a longer lever than the former: but, on the other hand, the mechanical action of the
former muscles to press the points V strongly into contact, by bringing XN B into a
straight line, receives very weak opposition from the latter muscles, which have already
brought ghB into a straight line; therefore the joint effect of these two pair of muscles
will be to press together and raise the points V.

Mechanism of the Larynx. 343

Magendie, to open the glottis by separating the arytenoids, but by
Sémmering and Bichat to close it. Other writers follow one or
other of these opinions, combining the different muscles after their
own fashions, without attempting to support their statements by
mechanical reasoning deduced from the structure and connexion
of the parts. In the following table I have brought together the
functions of the muscles according to the views I have taken

in the preceding pages.

CricoTHYROIDE! stretch the vocal ligaments

Antagonists.
—_——,

Govern the pitch

THYROARYTENOIDE! relax the vocal ligaments,
of the notes,

and place them in the
VOCALISING: POSIELOMa[s\- <is(e1selciele lens + aj is[a eines

fates tevin EOSIN! aaagodecodunoancorio as open the glottis )
3
3
=) (CRICOARYTENOIDEI LATERALEs press together the)
s front portion of
<
the Arytenoids. . Govern the aperture

together close of the glottis.

ARYTENOIDEI TRANSVERSI the glottis.

ET OBLIQUI press together the hin-
der portion of the
Arytenoids..... ee

Lest it should appear to some of my readers that I have, in
stating the uses and actions of the several parts of the Larynx,
expressed myself more decidedly than I ought to have done, I
beg to state that this decided style was adopted for the sake of
brevity, and that it is with the greatest deference that I have ven-
tured to offer opinions in many cases so different from those of
former writers. At the same time I have endeavoured to explain
the mechanical grounds of these opinions as clearly as_ possible,
and to distinguish carefully between those portions of my ex-
planation that rest on mechanical facts, and those that are merely
deduced from inductive reasoning.

ROBERT WILLIS.

344 Mr Wits on the

NOTES.

Note A.—Page 326.

M. Savart, whose labours in every branch of Acoustics have contri-
buted so greatly to the advancement of that science, has written an ingenious
Memoir, (Annales de Chimie, t. 30), in which he has endeavoured to shew
that the sounds of the Larynx are produced, not by the vibration of the
vocal ligaments, but in a manner analogous to those of the little instru-
ment called a duck whistle, of which he has given a theory with experiments ;
this machine consists of a small circular box, in the centers of the flat sides
of which are two holes exactly opposite to each other. When a current of
air passes through these holes a sound is produced, and his whole explana-
tion rests upon the analogy between the section of the Larynx (Fig. 1.)
and of this instrument, of which the glottis and pseudo-glottis are sup-
posed to represent the two holes, and the ventricles the cavity. But his
mode of obtaining the form of the laryngeal cavity is to take a cast of
it in plaster, by which the ventricles are of course distended, and made
to assume a magnitude and consequence which they never can possess during
life, but which are essential to his theory. Neither does it appear to me
that he has been successful in applying this explanation to the muscular
structure of the Larynx. This instrument had been before made use of
with great success by Kempelen for the explanation and imitation of the

whistling and hissing sounds of the human voice, (Vide Mech. de la Parole).

Mechanism of the Larynx. 345

These ventricles appear to have no use considered as cavities, but to
arise merely from the form of the lining of the Larynx, which, after se-
parating above the glottis to isolate the vocal ligaments, and leave them
free for vibration, again returns to form the pair of folds which constitute
the pseudo-glottis, and serve to protect the glottis from the accidental
intrusion of foreign bodies.

The most ordinary appellation of the vocal ligaments is vocal chords,
but this term, which implies an /solated vibrating ligament, ought certainly
to be abandoned as conveying a most erroneous notion of the structure of

the parts in question. They have also been termed the lips of the glottis.

Note B.— Page 328.
Let CD, AB, Fig. 22, be the longitudinal section of a tube, the trans-

verse section of which is a parallelogram, whose longest side is considerably
greater than its shortest, which is equal to AC.

Let this tube be terminated on its upper side by an elastic membrane
DE, attached on three sides to the tube, but having a free edge opposite B,
(similar to the membrane in Fig. 9).

Suppose the extreme position of this membrane in performing vibrations
to be DF and DG, and let a current of air be passing along the tube
in the direction of the arrow. Now when a membrane vibrates under these
circumstances its motion will be influenced by two causes.

First,—It is well known that when a current of air passes through a
diverging tube, such as ACDGB, that, by what is called the lateral com-
munication of motion, it gradually communicates its onward motion to the
particles of air which were at rest in E’DG, and carries them away with it
creating a rarefaction in H’DG, which occasions a superabundant pressure on
the outer surface of DG, by which it will be urged towards DE. This there-
fore acts as a retarding foree when the membrane is passing in the direction

EG, and as an accelerating force when it is moving in the opposite direction,

346 Mr Wittts on the

Second,—In a tube of the form ACDF'B the current exerts an out:
ward pressure, which will act on the membrane DF, accelerating it in its
passage from J towards HZ, and retarding it during its return to 7’. There
is an intermediate position DH, in which the current exerts no pressure
on the sides of the tube, and therefore none on the membrane.

We may easily conceive then that when the current was first admitted
into the tube, it might, from: the first cause, occasion a superabundant
pressure upon the outer surface of DE, which would set it in motion
towards DF. By virtue of this motion it would pass the line of equili-
brium DH, and would then be soon brought to rest by the resistance
arising from the second cause and its own elastic force. From this posi-
tion its elasticity and the pressure of the second cause would return it, it
would pass DH, be again brought to rest by the resistance of the first
cause, again return, and so on; in this way it would oscillate for some
time till the friction and resistance of the air, rigidity of the membrane, &c.
gradually reducing the extent of its vibrations, would bring it to rest in
the position DH.

Experiment shews however on the contrary that as long as the current
is maintained the vibrations continue. This may perhaps be explained by
a closer examination of the nature of the force arising from the lateral
communication of motion. When the membrane is passing from DF' to-
wards DG, and has got into the position DH, we suppose this phenomenon
to begin; but as the rarefaction it occasions proceeds from a motion gra-
dually imparted to the air, it is plain that it takes fime to perfect it, and
hence at any given point K the rarefaction is not so great when the
membrane passes it in going towards DG, as it is when the membrane

returns from DG.* Hence the retarding force at each point K in going

* This effect is assisted too by the circumstance that when the membrane is receding
from the current, the circumambient air can more easily rush in to supply the deficiency
than it can when the membrane is returning.

Mechanism of the Larynx. 347

outwards is less than the accelerating force at the same point in returning,
and the difference gives us a force to balance the loss from friction and
resistance, which will therefore keep the membrane in motion as long as
the current is kept up.

A similar explanation will apply to the case of the reed of an organ pipe,
to the free reeds now so much in vogue, and to every other case in which
a vibratory motion is maintained by a current. For instance, let 4 BCD
Fig. 21, be a transverse section of the plate of a free reed, and let EF’ be
the two extreme positions of the vibrating tongue which passes through
the aperture BC of the plate, the dotted line being its position of rest.
When _in the position #’, the current indicated by the arrows rarifies the
air above the tongue by the lateral communication of motion, which action
ceases the moment the plate gets to the level of BC, while a similar process
commences at the lower surface of the tongue, and ceases when it returns
to BC; affording in both cases a retarding force in going from BC less
than the accelerating force in returning to it, and therefore maintaining
the motion as long as the current is kept up. Here the tongue is first
started into motion by the upward pressure of the current; and if the
position of the tongue be not accurately adjusted, it is found that it will
either assume a position of rest a little above that which it takes when
no current acts upon it, or else will get very slowly into motion.

I propose to enter more fully however into this subject hereafter.

M. Biot* has attempted to explain the motion of an organ reed in a
way which would be perfectly satisfactory upon the hypothesis of perfect

elasticity and non-resistance of the air, but in no other case. Were his

* Physique, t. II. pp. 166, 172. Precis elementaire, t. I. pp. 429, 431. Also Pouillet.
Physique, t. II. p. 180.

M. Biot also adapted two lips of India rubber to a pipe connected with organ bellows,
and upon passing the current of air through them he obtained sounds. (Precis elementaire
de Physique, t. I. p. 462.)

Vol. IV. Part III. oy

348 Mr Wits on the

view of the action of the current correct, it is manifest that the reed would,
after a few oscillations, assume a position of rest in every case. These
remarks of mine have suggested to Professor Airy the investigation of an
elegant law, for which I must refer to his ingenious paper in the previ-
ous volume (p. 369) “ On Certain Conditions under which a Perpetual
Motion is possible.”

Note C.—Page 337.

The offices generally assigned to the cricoarytenoideus posticus are to
open the glottis by drawing the arytenoid backward, and to stretch the
vocal ligament. Now it is perfectly true that this muscle (NW Fig. 8)
in drawing the arytenoid from the mesial plane to open the glottis will
affect the tension of the vocal ligament 7'V by increasing the distance 7'V.
But the only function in which the tension of the ligament is concerned
is vocalisation, and for this a peculiar position of the ligament is required,
which is given by the thyroarytenoidei, while the complete and direct
regulation of the tension is also provided for by the joint action of the
cricothyroidei and thyroarytenoidei. On the other hand, the effect of the
cricoarytenoideus posticus upon the tension is very slight at the first
departure of the point V from the mesial plane, indirect and inconsiderable
in every case, and it cannot act without drawing the cartilages asunder
and out of the vocalising position; therefore I infer, that this muscle is
never concerned in adjusting the tension for vocalisation, and that its effect
upon it may therefore be neglected.

Again, the phrase “ drawing the arytenoid backward,” is a loose one,
and implies that the ligament BRBS is relaxed by this action, which
is by no means the.case. I have attempted to shew that this muscle
produces rotation round the axis OP; and as the bundle of ligaments
radiate from about that point B of the cricoid where the axis intersects
its surface, it is plain that the rotation of the arytenoid will scarcely affect

their tension. It is true that those fibres of the muscle which lie nearest

Mechanism of the Larynx. 349

the mesial plane are directed so as to draw the arytenoid towards B; but
this is counteracted by the fibres that lie farthest from the mesial plane;
and as we may assume that the whole of the fibres of the muscle act at
once, the resultant of their action will be found as nearly as possible
perpendicular to the axis of articulation OP.

Note D.—Page 336.

It is worth while to ascertain the state of tension of the vocal ligaments
when at rest, which we may readily infer from the application of the test
here described. If the finger be lodged in the space ED, Fig. 2, and a
bass note sounded, the larynx will descend from its position of rest, and
ED be enlarged; if a high note be sounded, the larynx will ascend from
its usual position, and the space HD be diminished; but an intermediate
note may be found, the sounding of which will not remove the larynx from
its ordinary position of rest, or alter the usual magnitude of HD, and this
note will be the average pitch of ordinary speech. Now, as I have shewn
that the space HD indicates the tension of the vocal ligaments, I infer
from this that in the position of rest these ligaments possess the tension
required for the average pitch of speech, requiring to be relaxed for deeper
notes, and stretched for higher. But as in this state of tension the breath
passes between them without being able to elicit vibrations from them, we
see the necessity of some such conditions as those I have described to

enable sounds to be produced at pleasure.

Note E'.—Page 338.

Some of the fibres of the arytenoideus transversus are attached to the
cricoid at one extremity, and are sometimes described as distinct muscles
under the name of arytenoidei obliqui; they conspire with the transverse
fibres in drawing together the hinder portion of the arytenoids, and by
their oblique direction assist the cricoarytenoidei laterales in depressing the

arytenoids.

“
“
ro

350 Mr WItLtIs on the

DESCRIPTION OF THE PLATES.

PLATE XXII.

Tur letters of reference belonging to the shaded figures on this Plate are con-
tained on its accompanying outline Plate, marked Plate 22*.

Fic. 2 is a section of the vocal mechanism in its natural state, made by a plane,
technically called the mesial plane, passing through the mouth, larynx, &c.
and dividing the head symmetrically. In the outline of this Figure the mouth,
chin, tongue, &c. are sketched in to make the relative situation of the Larynx
to these parts more clear; the shaded Figure is confined to the Larynx and
parts immediately adjacent.

MPA, upper part of windpipe.

MTOLIP, laryngeal cavity, opening at LJ into the pharynx.

trIL, part of the cavity of the pharynx, which is continued above ¢r
into the nostrils.

efghHk, cavity of the mouth.

a,b, lips. e,d, teeth. ef, palate. fg, soft palate. h, uvula.

kH, tongue. LO, epiglottis. IK, cesophagus.

TV, vocal ligament, constituting one side or lip of the glottis.,

TB, upper ligament, constituting one side of the pseudo-glottis; the dark
opening between these is that of the ventricle.

PQ, DM, cut edges of cricoid cartilage.

ETF, cut edge of thyroid cartilage.

8, cut edge of arytenoideus transversus.

Frc, 1 is a section made by a plane perpendicular to the former, and passing along
the line AB and BC in Fig. 2; looking towards JK.

The line GG passes through the vocal ligaments and glottis,
LL, through the superior ligaments and pseudo-glottis.
VV, through the two ventricles,

Mechanism of the Larynx. 351

Had this section passed nearer to EF (Fig. 2) these ventricles would have
appeared somewhat deeper, and considerably higher at their inward extremities.

Fic. 5 is a similar section, having the vocal ligaments in another position, and
Fig. 6 an enlarged diagram of part of these sections, which is sufficiently ex-
plained in the text.

In Fics. 3, 4, 7, 8 the Larynx is represented as removed from the surrounding
parts, and stripped of the epiglottis and investing mucous membrane, leaving
the bare cartilages, muscles, and ligaments; but still retaining in their pro-
per relative positions those parts which are left.

Fic. 3 is a section of the Larynx in this state, corresponding to Fig. 2; the cut
edges of the cartilages being therefore alike in these two figures.

EmCG, the thyroid cartilage. G, its upper horn. C, the place of its lower,
by which it is articulated to the cricoid.

AnBC, the cricoid cartilage.

F, the arytenoid cartilage.

EF, the vocal ligament.

FB, the bundle of ligaments uniting the arytenoid to the point B of the cricoid.

Emea, the thyroarytenoideus muscle.

Xe, the cricoarytenoideus lateralis,

s, the transyerse section of the arytenoideus transversus.

BL, the projection of the axis of articulation of the arytenoid with the
cricoid.

mn, the space between the thyroid and cricoid, which may be traced exter-
nally at DE (Fig. 2.)

Fic. 4, the external elevation of the half of the Larynx removed from Fig. 3.

EmcH, the thyroid cartilage. H, its upper horn. C, its lower horn, arti-
culated to the cricoid.

AnBC, the cricoid cartilage.

AK, the cricothyroideus muscle.

Fic. 7. A bird’s-eye view of the Larynx from above.

GEH, the thyroid cartilage embracing the ring of the cricoid rwXw,
and capable of turning on the axis wz, which passes through the lower horns
C, Figs. 3, 4.

NF, NF, the arytenoid cartilages.

TV, TV, the vocal ligaments.

NX, the right cricoarytenoideus lateralis, the left is removed.

352 Mr Wits on the Mechanism of the Larynx.

Vicf, the left thyroarytenoideus, the right is removed.

Ni, Nl, cricoarytenoidei postici.

The arytenoideus transversus is removed.

B, B, the ligaments uniting the arytenoid and cricoid.
Fic. 8 is part of Fig. 7 enlarged, to shew the direction of the muscular forces

which act on the arytenoid cartilage.

QNVS, the right arytenoid.

TV, its vocal ligament.

BRS, the bundle of ligaments uniting it to the cricoid.

OP, the projection of its axis of articulation.

hg the direction of the force of the thyroarytenoideus.

NX cricoarytenoideus lateralis.
NW cricoarytenoideus posticus.
NY arytenoideus transversus.

PLatTE XXIII.

Consists of figures of apparatus and diagrams which are sufficiently explained in
the text.

ERRATA.

Page 333, line 12 from bottom, for Figs. 7, 8, read Figs. 3, 7.
—— 338, — 6 from top, after “ the point S$,” insert (Fig. 8). *

bray Mio Ty 5 y ° , Ts: ane = = o

o
28 TLL TOA JOS WHE WED FHL JO SNOLLIVSNFAE- a

Lc alee ere aes |;

Transactions of the Cambridge Phil Soe Vol.4 Pl.23.

IN. mE
G
22 K
———— Kg
aH
S --
—s Fr
— =. = Seas =
, n

None Calle. 382 Siren

XIII. On the Inverse Method of Definite Integrals,
with Physical Applications.

By THe Rev. R. MURPHY, B.A.

FELLOW OF CAIUS COLLEGE,

AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read March 5, 1832.]

INTRODUCTION.

Tue mass of theory, on the subject of Definite Integrals, con-
tributed by Fourier, Poisson, Cauchy, Gauss, and other modern
analysts, is very considerable. Many of their properties, and
applications have been pointed out, and the calculation of their
numerical values facilitated. We may, therefore, regard the
direct calculus of Definite Integrals as already formed, to a
certain extent.

But as subjects are generally best understood, when examined
on all sides, it would be advantageous, even in this point of
view, to possess an inverse method of Definite Integrals, by
which we may re-ascend from the known integral, to the un-
known function under the sign of definite integration. At the
same time objects of importance in the pure, and physical mathe-
matics would be attained. Euler and Laplace have valued the

354 Mr Murpnuy on the Inverse Method of

interpolated differential coefficients of fractional orders, for such
functions as may be simply represented by Definite Integrals of
a peculiar form. If we extend this principle to all functions of
operation of the distributive kind*, (that is, such whose action
on the whole, is the sum of the actions on the parts), and
if we can represent a given algebraical function by a Definite
Integral of the proper form, this view will be complete.

Again, the phzwnomena of the physical sciences generally re-
sult from an infinite number of the elementary actions of the
particles forming the system under consideration. Such an inverse
calculus would conduct us from the observed phznomenon, to
the laws of the elementary actions.

The following researches have been conducted, with this object,
chiefly, in view. The analysis consists of two parts, corresponding
to two distinct classes of phenomena in nature. Namely, such
as result from sensible, or developed powers, the action of which
is generally insensible, at infinitely great distances; and such as
are referred to latent, or. neutralized powers, which only become
sensible at indefinitely small distances from the component par-
ticles of the system, whence the actions emanate. Of the first
part, which I now lay before the Society, the following is a brief
abstract.

Adopting 0 and 1 throughout as the limits of integration,
there is a character common to all the integrals of the functions
commonly received in analysis; if the function be multiplied
by #, that is, any positive power of the variable, the integral of
the product, which is a function of 2 converges to 0, for infinitely
great values of x. To revert from the function outside, to the

* Annales de Math. Tom. V. Servois’ paper; Vid. also Tom. III. Francais’ paper.

Definite Integrals, with Physical Applications. 355

function under the definite integral, when the former is a rational
function, we have only to multiply it by ¢* and taking the

=a bea ng ‘ =
coefficient of zim the product, divide it by ¢ This theorem, proved

in Section (1), is there also extended to any number of variables;
the remaining part of the Section illustrates the nature of the
application of the theorem, and includes the mode of treating
irrational functions.

The functions considered in the first Section are all continuous,
the mode of treating discontinuous functions is shewn in the
second; it was necessary to attend to this class, because the
phenomena presented by nature are mostly of that kind. Thus
the action of developed electricity follows two different laws,
according as the point acted on, is within or without the surface
of the body; the function which expresses this action generally,
must therefore be discontinuous, the interruption taking place
at the surface. In like manner heat is propagated simultaneously,
in the earth, the sea, and the atmosphere, but according to dif-
ferent laws in each, and other instances of discontinuity are
observable in every department of physics.

Now we shall obviate the difficulty thus presented by discon-
tinuous functions, if we can obtain a formula, which without
any alteration of form may continue to represent the function,
under all circumstances. This object is attained by a simple
application of the theory of Algebraic Equations. For when an
Equation is resolved by the method given in my former paper*,
the expression for the least root is in a form symmetrical with
respect to all the roots. If for simplicity we only consider two

* Cambridge Philosophical Transactions, Vol. IV.
Vol. 1V. Part Ill. Zz

356 » Mr Mourpuy ‘on the Inverse Method of

roots,";one’ a constant, the other variable, commencing from
nothing and‘ imereasing indefinitely; it is evident that the above
formula,. will. represent. the. variable root, as long as it is less
than the constant; but the instant the variable exceeds the con-
stant root, the formula will cease to represent the former, and
from thenceforward it will express the constant. This principle
when fully developed, meets every case of discontinuity.

In the third Section, the preceding theory is applied, to the
phenomena of developed electricity, from thence deducing the
law of accumulation on the surface. The function which ex-
presses the action of a closed conducting surface, charged in any
conceivable manner with electricity is discontinuous, but the
parts are not independent in consequence of the known law of
force to each particle at different distances. By the principles
of the first two Sections, we may, by observing the law of action
of the electrised body, deduce the law of distribution on its
surface. When for instance the action of a sphere, in any manner
electrised is observed at distances greater than the radius to vary
inversely as the (~+2) power of the distance, the law of accu-
mulation, which is expressed by a differential coefficient of the
n” order, is of a remarkable kind. There will then be ~ nodal
or transition lines on the surface, in which the electricity remains
unresolved, and which divide the sphere into ~+1 portions con-
taining alternately the positive and negative electricities; in each
portion there is a line of greatest accumulation (which becomes
a point in the two extreme portions); the transition lines, and
lines of greatest accumulation, divide the surface into belts, con-
taining alternately in pairs, exactly equal quantities of the op-
posite fluids. And by the superposition of several systems of this
kind, all the phenomena of developed electricity are produced.

Definite Integrals, with Physical- Applications. 357

When a sphere, already electrised, is subjected to the influence
of an electrical point without it, the law of accumulation is simple ;
for the quantity of fluid which must be superposed to that already
accumulated on each annulus, to make it every where equal to
a certain constant, is always inversely proportional to the cube
of the distance from the influencing point*.

As the application to the theory of developed electricity, offered
a sufficient illustration to the principles of this part; at the same
time that it conducted to new and remarkable results, it seemed
unnecessary, considering the convenient limits of this paper, to
insert other physical applications which I have made, particularly
on the subjects of heat and magnetism.

* The electrical action in the third Section, is measured by the tension of the fluid
which would be produced in an infinitely thin rod, communicating with the electrical
body, by the attraction or repulsion of the latter; it is what Mr Green, of Nottingham,
in his ingenious Essay on this subject, has denominated the Potential Function.

ZZ 2

358 Mr Murpny on the Inverse Method of

SECTION I.

PRINCIPLES RELATIVE TO CONTINUOUS FUNCTIONS.

(1) Preliminary Observations.

1. For the purpose of uniformity, and simplicity, the integral
limits which we take are 0 and 1*. If ¢ be any variable with
respect to which integrations are performed on any function, and
the limits of ¢ are the finite quantities @ and 4; then if we make
t=a+(b—a)t’ the corresponding limits of the new variable, ¢’ are
0 and1. But if the limits of ¢ are one finite and the other infinite,

as a and +, put then ¢=a+ ae if they are both infinite as

1-2?’
—o and + make Ser aayi generally, therefore, by these

substitutions, and by more convenient ones in particular cases, we
may always make 0 and 1 to be the integral limits.

2, Let f(t) be any function of ¢, and make
PO)=ffS), PAN=hSOH-46 P(Q)=LS(O-#, &e.

none of which definite integrals is supposed infinite. Then, «
being any positive integer, we have $(2)=//(é).¢. Put 1-¢ for
t (the limits of ¢ are also 0 and 1, changing the sign of the
integral) and let the successive finite differences of $(x) when «

* The limits uniformly adopted by Gauss. ‘“ Methodus Nova Integr.”

Definite Integrals, with Physical Applications. 359

is made =0, be represented by A.(0), A*.¢(0), &e. The equation
when both its members are expanded becomes

x.(x—1)

iF (t).t* = (0) +a.Ap(0)+ 13

-A’6(0), &e.

x.(@—1)

= -fkf (+ efp ft) t — a or eee hf (t).t° + &e.

and making x successively =0, 1, 2, 3, &c. we see that the cor-
responding terms in both series affected with their proper signs
are exactly equal, the two series are therefore identical for all
values of «. But the values of x from —1 to ~” are rejected
as, i general, giving infinite integrals, for then if A, be the
absolute term in f(¢) it is evident that fA,t* will be infinite. In
the particular case where J(t) contains no positive power of ¢
below ¢", we may then include the negative values of x, between
0 and —m. But in all cases when P(0), P(1), (2), &e. are
finite, we may assign any value to 2 between —1 and 0, or be-
tween 0 and +0”, and always have p(x) =f f(t).t°; we shall
therefore in the following theory attribute to x all values included
between —1 and +.” and suppose $(x) to remain finite, during
that interval.

3. Let us next consider the value of (x) when w=, and
J (t) is any of the functions commonly received in analysis.

If f(4) always remains finite between the extreme values ¢=0
and ¢=1, let the greatest value of J (t) be represented by 4 and
the least by 4’, then ?(t)=f f(d).t? is included between {4.7

ba A glen A :
and /4’.t, or between ery and za It is therefore =0 when

x is infinite.

360 Mr Morpny on the Inverse. Method of
If f(é, in the same interval, becomes infinite for a particular
value of ¢ (suppose when ¢=a), it is then of the form —

being a finite function of ¢, of which the greatest and least values

may be represented by p and p’; ¢(«) then is between

a ? (Ae!
PL Gaps 24 # Lea

Now since

s 1 1 1 m 1 1
Map = 341 ia we aH * (@4).@e+2) " fama “Car is

it is obvious that when «=, $(x)=0 in this case also, with the
exceptions which we are now about to examine.

When a=0, then ¢(«)= fi Pts fP.t-", and putting for Pits

extreme values, (+) is evidently included between ——7—| and
aes and is therefore nothing, when 2 is infinite; but when

m is not less than unity, it is clear that ¢(0)= J 2 is infinite, the

part of P not involving ¢, giving in this expression an infinite
integral ; this case is therefore inadmissible. (Art. 2.)

When a=1, then ¢() is included between
# f # ‘
if therefore m be not <1, $(0) is included between

1 ae
P SGay and: p lay

Definite Integrals, with Physical Applications. 361

and is infinite, the absolute term in P giving as above, an infinite
itegral; rejecting therefore, this case, and supposing m <1, if we
integrate by parts, we find

f ee: x 4 ta ,

w(@—1)" we +(1—m)* Je (¢-1)"’

this integral therefore converges to 0, as 2 increases to infinity.
Suppose next that (4) becomes infinite for several values of ¢,

as t=a, t=a, t=a,, &c.,

P
(¢—a)".(¢-a@)" (¢—a@’)\".....

then f(¢) is of the form

which being resolved into simple fractions, the same reasoning as
above shews that for all the admissible cases, we have ¢(«)=0
when «=a.

Lastly, let f(é) be imaginary, so that /(¢)=P+@Q./—1, P and Q
being real functions of ¢, then ¢(2)=/{,P,t'+./—-1/Q¢t, and by
the above reasoning, both the latter functions vanish, and conse-
quently ~(z) also =0, when «=a,

4. From this examination we see that the inverse problem
“to recur from (2) to f(t)” consists of two parts*, first, when
(x) converges to 0 as a increases to «©, and is therefore essentially
composed of negative powers of x; in this case which forms the
subject of Part I. /(4 is of the form of the usual functions re-
ceived in analysis; secondly, when ~(«) does not consist of such
powers, for instance when it is always zero, and then /(¢) belongs
to a new and remarkable class of functions which will be treated

* This is the division of Integrals into large classes, alluded to in p. 439, Vol. III.

Camb. Trans.

362 Mr Mourpny on the Inverse Method of

of in Part II, and it has been stated in the Introduction, to
what classes of natural phenomena, the respective parts corres-

pond, It is obvious that the form of ¢(x) when rational is

= + z + C + &e. in Part I. We shall now proceed to the prin-

ciples, by which we may revert from the given function ¢(x) to
the unknown function /(é),

(2) Inverse Method, for Rational Functions.

5. When the known function (x) is rational, seek the co-
efficient of : in (x).t-*; dividing it by t, the quotient will be
the required function jt).

To prove this, suppose @ to be any negative number; the co-

efficient of in ¢(«).¢-*, which we shall represent by 7' is also

the coefficient of * in ¢(a).t-°. We shall get by actual integration

from ¢=0 to ¢=1, the equation,

eee eereeeeee

(a) 4.-1_ 2)
(ae = Ee eee(l),

and taking the coefficients of * at both sides

«w—a
Now, by supposition, ¢(«) is of the form 4 + = + c + &e.

Consequently,

tha [dB Sone) fhe SoS ote)

Definite Integrals, with Physical Applications. 363

The coefficient therefore of

feo af. BC
Ae eaaheiah atak Ge Se
= (2).
Substituting this in equation (2) we have [4-6 =9() cdasth see (3)

which shews that /()= = as announced above.

6. The following examples are intended to shew the manner,
in which this theorem may be applied.

1
x+m’

Ex. 1. Given {,f(é).t¢= m being essentially positive,
(vide Art. 2), to find /(¢).
1

: i 16,99
We must find the coefficient oe in ¢ SS

—z

Now since ——
r+m

x aa
= {12h 94+ -.1.0- 133 > 3° (ho k7)° +&e,|
1 mm mm &
x { phage ae,
the coefficient of

Sh tims m* 4
zn aap =1tmhL (jt Pohl. che + Doig: (h. 1. d+ &e.

=",

therefore the required function /(¢)=

Ex. 2. Given {//(d).t°= le+my to find f(¢), (x being integer).

Vol. 1V. Part III. 3A

364 Mr Mourpny on the Inverse Method of

¢-*

Since... +00 (@tmy

={l- —ehlj+s, @ (he 1. .t—&e.}

x fa-"— n.man*= + MOTD) mte--*— &e.},

(1.3) m S ; ‘ye

the coefficient of 4 -{l+mh.1.(é)+

a 1.2.3...(2—1)

e. (1.7)

» Lito Bpla—a):

roan

| eae):

+Ke.}

and therefore in this case /(é)

Ex. 3. Given {,f(¢).¢*=h. 1. (1+ —_), m being essentially posi-

tive, and a either positive or between 0 and —m.

In this case we have

rebel. (1+ 2) =e. fet mh.1 (9+ St" (hry ee}

a 1
x{ = 2° era +&el,
and if we select the coefficient of . from the products of the
corresponding terms in both series (which are the only products

that contain =) we get, the coefficient of ae in

-* GN ae sje oe
t hl (1+—2—) =e. fat oh + S53 -(h- bo? &e.3

and therefore /(¢)=é"-'..——~

Definite Integrals, with Physical Applications. 365

(3) Means of facilitating the Calculus of f(t).

7. By this method we may always revert from $(«) to /(¢);
but to facilitate the process, it will be found convenient to reduce
(x) to its simplest form, previous to the application of the
general theorem.

Thus suppose ¢ (x) expanded according to the descending powers
of «+m (m being any positive quantity) i. e. if

A, A, A,
P= stm + emp t tmp

+&e.

then assuming the result of the second Example above, we get,

A,. (1.2) 4,.(h.L 7),

a ee ee - se 2
ACE

f=" {4,4 A hil. (7) +

8. When ¢(z) is a rational fraction, the highest power of «x
in the denominator being greater than that in the numerator,
and the denominator not vanishing in the interval from «=0
to »%, then if we decompose ¢(2) into its simple fractions, the
results of the first and second Examples above will give us the
value of f(¢).

Ex. 4. Given
_ _#.(@—1).(@— 2)... (v—n+1) |
()= G51) @ +2) @+3) Bast (cx+n+1)’
to find f(z). Put
N, N: N+
ares tars i Es

which compared with the above, gives
a.(a—1).(a—2)...(a—n +1)=N, (a + 2) (aw + 38)...(u + +1)
+ Nz (a +1) (a + 3)...(v +” +1) + &e.
3A

366 Mr Morpny on the Inverse Method of

and to determine M,, N. &c. suppose, successively, x= —1, —2, —3,
&c. thus we obtain

So 14g Re RE ae n+1- n. — n.(n—1) (n+1).(m+2)
Ni=(-1); N=-(-1).5--: Me(-19. GB. tke.
and therefore

e@)-- fat at ge

“+2 heise 1.2 “2+3

—&e.} ;

and by Ex. 2 we have therefore

n n+ na.(m—1) (n+1).
J (tj) =(-1Y a5 I ee, =p

\
~
w9/5
aE
to
_—
~
S
lo)
os

an elegant result, to which we shall hereafter have occasion to
refer.

; 1
Ex. 5. Given ¢(«)= (@+h).(@+2h)......(@+2A)

Applying the same method we get in this case

® PS. (1 =
f= 2302. (a 1),8 7)

9. Let us next consider logarithmic functions, beginning with
h. LZ («), supposing that neither P nor Q vanishes from x =0
to ©, and that the highest powers in the rational functions
P and Q may be equal, and be multiplied by equal coefficients;
which conditions are evidently necessary to make ¢(x) remain
finite for all positive values of x, and to vanish when t=.

To find /(é, let P and Q be resolved into their simple frac-
tions, so that

P=C\at+ a) (a+a,)......(a+a,),
and Q=C.(a+;).(v+b,)...... (a +6,) ;

Definite Integrals, with Physical Applications. 367

P a—b, a, — b,
therefore h.1. Qo h.1. (a + ss, +h. 1. (1 af ara

) + &e. =o (2).
Hence by Ex. 3,
f(t)= Tanne LOO + th) (4 Oh ts)?

For other instances of Logarithmic functions, we refer to
Note (A) at the end of this paper.

10. Another form of ¢(z) to be considered, as frequently oc-
curring, is that of fractions of which the numerator and deno-
minator are composed of a variable number of factors; the
relation between P(x) and $(«+1) will enable us to transform
such a function into a series of simple fractions; we may then
recur from ¢(x) to /(¢) as before.

24 6..02609

Ex. 6. Given $(x)= a CE

= 1)’ to find / (2).

By the nature of this function we have

(2a +2) p(x) =(24+3).p (41),

Substitute for p(x) the series

A B C

—— + be:
ein “2in4l] | eines e

the equation becomes by actual division,

24(1—n) | 2B(l-n-1) | 2C.(1—n—2)
rt+n “W evin+l A 2+n2+2 Ke.
24.(3 —n—1) 2B. (5 -n-2)

oa fo —_—<—<—<<—_——_ £ra
r+n+1 % X+nN+2 &

368 Mr Murpny on the Inverse Method of

and putting »=1 to make both series comparable, term by term,
we get

OB AO fOD agen Ae, BB eee
+2 | #48 gf 948 oes tA?

which gives B=4.5, caB.2, Bele? ke.

so that $()=4 | qohqar eight él

+1 ' 2° 242" 24° sya tke};

f= A 4 Ltt ES 4&e.3 = rent

the constant 4 is found by giving x a particular value as 1,

=
ty Papo?
which gives 4=}.

Ex. 7. Given

a.(a+a).(a+2@)...... fa+(x—1).a}

Following the same steps, we find here

_ penal) co 2 (ba)
whence f(t) = Ata aes

A similar method is to be used, when the relation between
p(x) and @(x+1) is of a more complicated nature, or when that
relation refers to more than two such functions.

11. The denominators of the simple fractions in the two
preceding articles have been real; such quadratic factors as would

Definite Integrals, with Physical Applications. 369

introduce imaginary quantities, when decomposed into simple

factors, may more conveniently be left in the quadratic form,
as in the following example.

7 : _ Ar+B
Ex. 8. Given ¢(2)= Wate
It is easily seen that

1 B° B
o)= 4A} Gap t Gray 84

1 & B
+(B— Ae) ery eal * Gray ~*}

and therefore by Ex. 2.

Fara tere eee
J (t)=At*. f1- Aber) + PaO:

1 1)
(ashe 14> ge? (oe
QeaAidyfM Plt) i) al

1 1.2.3

=t-'{4. cs (ah.1.}) + BAAS sin (ab.1.7)

— a 2
= ra ) and y=tan~’. ate,

put »?=A’+ (

Hence f(é)=¢7'.usin (y+Bh.1. 2):

which result if differentiated » times with respect to 6° as inde-

pendent variable, will solve the case of 2

equal quadratic
factors.

370 Mr Mourpny on the Inverse Method of

(4) Inverse Method for Irrational Functions, and Functions of
several Variables.

12. None of the forms of ¢(«) hitherto taken, have involved
irrational quantities, but a simple modification of the preceding
results, will make them include surds. By Ex. 2,

1 n—-1
a (hl
“~ 1
if $(@)= Game then SQ)= eee

and to make this apply also to the fractional values of x, we
have only to put the denominator under its more general form

f(b. we

To prove this put ¢*"=7, the limits of 7 are the same as

those of ¢, af id
and jé""'. (ho. *) = Se) (he. =) a

but (hu. te _ f (bul. sli

om? (hu. \

a to find / (2).

If we expand ¢(), the form of the general term, abstracting
be Dates) aie ix
1.2.3...2.(@ + m)"**°

from whence it is obvious that /(¢)=

*Ex. 9. Given ¢ (x)=

from the sign, is

* Laplace, Theor. des Prob. p..97. Ed. 1820.

To make the limits 0 and 1 in Laplace’s formula, put e-*=t; and change the sign
of the integral.

Definite Integrals, with Physical Applications. 371

The corresponding term in /(é), is

1\"-3 _ 1.3.5...(2—1) en A leSc0e Ao)
But f(b-1.3) a SN) iw Ve ee,

thus the general term of /(¢), abstracting from the sign, becomes

pia (2a). (bh. L z)

r-*eos. 24 J hil. jh
A a. (7)

13. In applying the above principles for recurring from ¢ (2)
to f(t) in the equation /f,f(t).A=(«), we should attend to the
extent of the values to which 2 is limited.

from whence /(¢)=

When ¢(x) remains finite from «=0 to = we may assign to
x any value from «=—1 to x=, (Art. 2).

When ¢(x) remains finite from x=h to =~, i.e. if 4 be the

5 1 5
greatest real root of the equation peo we may assign to «

any value from «=h—1 to x=. For the equation may be put
under the form {,f(¢).t’.¢7-"=@(x); put #a—h=a’, and /(¢).t'=F(%),
J. f(b.’ = (a +h), which remaining finite from 2 =0 to a=
we may assign to «’ all values from 2 =—1 to a=+, i.e. we
may put for « any number between 4-1 and +,

Vol. IV. Part III, 3B

Mr Murpny on the Inverse Method of

372
When ¢(x) remains finite for all real values of x, we may in
the applications give « any value from —* to +«, and may

then use formula of the form

IF O(CLE\=$ @) £9 (2).

If p(x) remains finite for all values of 2 from x=—é to «=6,

then putting ¢=¢" and a= = the equation becomes
mn werd x
RSE) Ee ve =9 (=),

which integral remains finite from «’=—xb to x =nb, and if we
increase » indefinitely these limits tend to —» and +2, and

therefore we may use the formula
SGC VES GE tf sao) sa (sr)

for all values of 2’ between —2”d and +2b, or which is the same,

we may use
LS OE £0) =9 (2) + O(-2),

for such values of « as lie between —S and +6.

* Ex. 10. Given

ff. {e—i-y= 5 tan (F-) , to find /(?).
is T T 1 1 1 1
Since 9 tan (F2)= foo Riney saan o35 Xe
= (x) —(—2)..-putting p(a)=-L + a +&e.}.,

and it remains finite from «=—1 to +1, then deducing /(¢) from

o(«) by the usual method, we have
f= (1404+ ej =a.

. Formula 115.

* Vide Cauchy’s paper, Annales de Math. Tom. XVII

Definite Integrals, with Physical Applications. 373

When the highest negative power which enters /(¢) is ¢-", then
if (tt (with the limits of ¢ 0 and 1), is only finite when «>(h—-1),
but if « be <h-—1, the parts of the integral which then become
infinite retain their former finite values by making the limits of ¢
to be ~ and 1; and all the principles established in this section
will apply to them with the new limits.

14. Let f now denote a function of any number of variables

t, t, t’ &c. and ¢, a function of other variables 2, 2’, x” &c. under
the same restrictions, with respect to each of the variables as
before, and let all the limits of integration be 0 and 1; then if

there is given the equation
Rife feeccce fOwt BiiSid,
we may revert from ¢ to f, by the following theorem:

““Take the coefficient of

yw! yt

-7

yey a mee

fem
U2 20 &e. eeeeee

and divide it by ¢#?’...&c.; the quotient will be”

For let ¢.¢’ be the coefficient of = ine p.t—

y " ik . payne I —r ,-2
PMN rave eeactgtdes terete es in ar eae. yin g.t or of Tin pt t's
and so on, then (Art. 5) {¢'.t°=¢,
fie t =$, +. fhe .et’ =¢;
and the same reasoning evidently holds for any number of va-
riables.

It will not be necessary to annex any examples, to elucidate
the applications of this theorem, as they are similar to the ex-
amples for functions of only one variable. (Vid. Note B.)

332

374 Mr Murpny on the Inverse Method of

SECTION II.

PRINCIPLES RELATIVE TO DISCONTINUOUS FUNCTIONS.

(1) Method of representing Discontinuous Functions of only
one Break.

15. To bestow on the principles of this analysis, all the gene-
rality, required by physical problems, we must extend it to such
cases of definite integrals, as undergo a total change in their
values, under given circumstances. The attraction of a spherical
shell for instance, on any point is a definite integral. But when
the thickness is indefinitely small and uniform, and each particle
attracts by a force varying inversely as the square of the distance,
this definite integral is 0 when the attracted particle is within,
and a finite quantity when without the shell. The law of at-
traction is therefore interrupted when the attracted point arrives
at the surface of the shell. When the shell possesses a finite
thickness there are two breaks in the attraction, or the definite
integral by which it is expressed. The attractions of spheroids
of every form are similarly discontinuous, when the force follows
such a law that it becomes insensible at great distances. So also
in heated bodies; the law of expansion is different for the same
body, in the different states of solid, fluid, and gaseous. The
electrical phenomena present several similar instances, To ob-
viate the difficulty thus offered, we must seek an analytical
expression, which without altering its form, may continue to
represent the true value of the definite integral, or discontinuous
function, under all circumstances.

Definite Integrals, with Physical Applications. 375

16. To find a formula, which shall always represent the Jeast
of two quantities a and 8, abstracting from the sign.

It has been shewn, in the paper on the Resolution of Equa-
tions*, that if @(#) be any function of x involving only positive

and integer powers of x, the coefficient of : in aa. is the
same root of the equation ¢(x)=0 as that given by Lagrange’s
theorem, that is, the least. Hence it follows that if we seek the
ee or in h.l. (1+ oo , it will

represent the least of the quantities « or 8. Expanding, there-

coefficient of : in h.l.

fore, this logarithm, and taking the coeflicient of = in each term.
we get the equation,

(aBy? 1.1.3 («By

anes ene

The least of He = ale + Jul + &e.

il
B a+pB 2.4 °

17. Put = for a and 7 for 6B; we find thus:
: |
eed lee - “aie 1.1.3 _(aA)’
The least ae ar rare ea)’ + eae: (exey +e.
2 2

representing this series by |S, it follows that

LS a
Sy-e....0-)de a" when a>8,

and =0 when a<B,
: . 1 :
that is, we have a formula which represents 0 or — according

as a Is < or > BP.

* Camb. Trans. Vol. IV. Part I.

376 Mr Murpny on the Inverse Method of

Thus if a represent the distance of any point, from the centre
of a spherical shell, the formula which represents the attraction

both within and without (i. e. 0 and =| is — oe
a pa

18. To find a formula which shall represent /(2) when «<A,
and ,f(@) when a is greater than 6; f(a) being supposed to consist
of positive powers of a.

It appears from the paper above referred to, that the object
will be answered by taking the coefficient of : in —,f’ (x) h. 1 2)

x“

where f’ (x) is the derived function of f(x) and ¢ (2) =(#—-a).(«— 8).

Thus, if /(«)=a", m being positive, we must take the coefficient

of = in —mh.1. (1- a. that is, the term independent of x in
x a(at+ B)

ee ee eo ee

which evidently is

(a)” (m) (aQ)27" (m+8).(m) (aB)"*?
(eeart L (+e? 1.2 a ee

the general term of which is

m.(m+n+1).(m+n+2)...(m+2n—1) (aB)"*"

1.2.3...n ‘(a+ pyre
Cor. Putting = for a, and - for 8; the following formula will
1 1 :
represent ee when «>, and Be when >a, viz.

1 m ap m.(m+3) (aB)?
1

G+Ay 1 '(atppr 12 arp ke}.

Definite Integrals, with Physical Applications. 377

19. To find a formula, of which the value is ie when

a>B, and when 8>a; # being any quantity less than unity.

les
B—-—ha
1

oe 7 and

The order, in point of magnitude, of the quantities

1 . Mi 1 1
A-ha’ '8 the same as the order of - and a: For suppose a<B

then a+ha<8+h; and, subtracting (a+) from both sides, we

have a—hB<f-ha, attributing to 2 any value between 0 and B

again since h<f we also have Bh+ah<f+a; subtract ah+a from

both sides, .. Bk—a<f-—ah, attributing to 2 now any value be-
tween 3 and 1; whence it is evident that for any value of 4 from

0 to 1, we always have a—{h (abstracting from its sign) </—«h,

when a<f: and therefore if ~ bei —-.or <3 accordingly shall

AU A Aba eee :
a— bh at Raa | ~ Ge ake

Now by Art. 17, we have

1
.Ja-Bh 1
the least of | Cesar
B-ah
1 12° faB.(1+hy—h.(at+ By} 1.3 1 2" faB.(1+h)'-h.(at Bt, g |
1? ao that ereny sara: (a+ BY.(1—hy inf

which is obtained by putting a—£h for a, and B—ah for B in the
article referred to.

This series may be easily arranged in the more convenient
form,

378 Mr Morpny on the Inverse Method of

A Llp Mab 118 p 2B
Bu at 3 “Grpy t 24 e Fa a ep

where B,=(1 +h)”

x{ 1 1 (Q2n+1).Vh 1 (Q2n+1).(2n+3).2*h? 1
(Ay n+1 2(1—h)* tn +2 24.(1—-h't> ‘'n+3

a (1+ h)™.0 S a
= ape from ¢=0 to f=1.

20. To find a formula, which shall represent = when «a>,
and == when 6>a.

Suppose that H, is the coefficient of 2” in B,, in the last
Article, it follows that

ab 1.3 93 (aB)*
iT... 3+ 5h. Ga+py * 2.4 Gap + &e.
le
is the coefficient of ” in the least of ey , 1. e. it represents

B-ha

a >
OF Baxi according as «> or < f.

Integrating by parts, and observing that B,=the least of

1
aoe yy Rai SP rae AES
[ates oe get B= —le yay + Gani). a3).
—h

2n.(2n—2) (L+h)

* @n—1).(@n—3).Qn—5)* 2 + &e.}

2.4.6...2n (1+h)™
1.3.5...(2n—1)° Oh

Definite Integrals, with Physical Applications. 379

Comparing this expression with the series for B, in the last
Article, with which it ought to be identical; we may simplify, by
expanding each term in that series according to the powers of h,
and reject all powers higher than 2%", since they must mutually
destroy each other; or we may expand each term in the latter
expression, and reject the negative powers of h for the same

reason.

Adopting the latter process, it is obvious that the part of B,
between the brackets, is of the form

dee: ae

A,+A,h+ Ah? + &e.
. Got aa i2arae

2.4.6...2n

1
1.3.5...(2n—1) "9

and the part without =
2n.(Qn—1)...(m+1) | Wn.(Qn—1)...(n+2) 2n.(2n—1)...(m+3) ,,
1.2...0 SR CE err Re &e.}
An.(2n—1)...(~+2) 1 Qn.(Qn—1)...(m+3) 1 i
[+ (Oey ack? 1 Aeeeieey Oe

but since the negative powers of must disappear, this con-
sideration gives

1 2.4.6...2n Qn.(Qn—1)...(n+2)

Ao= 9% T3.5..(an—1)° 1.2...n—1
i 2 1 | 2.4.6...20 Qn.(Q2n—1)...(% +3) Rees

9% 1.3.5...(2n—1) 1.2...(m— 2)

substituting these values for 4, 4, &c. in the other terms, and
making the proper reductions, we get
1 3n 5n.(n—1) he
= — + ————__ .h +. ———_——__ ht’ &e.
- n+l + +l).(n+2) * (+l).(n+2).(n +3) E
Vol. IV. Part III. gic

380 Mr Murpuy on the Inverse Method of
and taking the coefficient of h" we have

_ (Qm+1).n.(n—1)...(a—m+1) |

5 (n+1).(a+2)...(u+m +1)
and therefore
Rf i i _ (Q2m+1).m.(m—1)...1_,
ES ees VEL, Oe Tea tg OR (m +1).(m + 2)...(2m +1)’
es (m + 1)° 5 §(m+1).(m+2)}*
Ha n= ™"1.(Qm +2)’ Huss Lm. 9 (8m +2) (2m +3) =
The required formula therefore is
1.2.3...m aire Q°"(a 3)”
(m+1).(m+2)...2m| 2.4...2m “(a+ f)"n?
1.3...(2m+1) (m+1)? Q°"+?(aB)n**
2.4...(2m +2)" (2m+3)° (a+ f)"*?
1.3.5...(8m+3) — {(m+1).(m+2)j* _— Qmt*(apyn** Par !
" 2.4.6,..(2m +4) °1.2.(2m +2).(Q2m+3)° (a+ B)e"** rie

(2) Method of representing Discontinuous Functions of any

number of Breaks.

21. These particular instances have been put under forms best
adapted for many of their applications. We shall now proceed
to more general principles, for the representation of discontinuous
functions.

To find a formula which shall represent
—i (a, B, Tyaees)s 2 (a, B, Yoe0e)s Ps (a, B, Ys-00) &e:

according as a, B, y &c. are respectively least.

Definite Integrals, with Physical Applications. 381

Denote by ¢, ¢., ¢: &c. the preceding functions, and let the

wee eal OCs ae be found
x a

ae des
coefficient of 7 im h.1. and be re-

presented by |S, then shall

ds ds dS
poe + d.. dp + ps. dy + &e.

be the required formula.

For when a is the least of the quantities a, B, y &c. then
S simply represents a, and therefore

qs_, d8_, as

dp ’ dy tau &e.

and the above formula reduces itself to ¢..
Similarly, when # is least, it becomes = ¢:; and so on.

22. To find a formula, representing a discontinuous function
¢~. which assumes the successive values ¢:, ¢, ¢:, &c. according
as a variable quantity *, commencing with a value=a, flows
through the successively increasing magnitudes 6, y, 6, &e.

Denote by S (a,z) the coefficient of + in h.1. era)

&e. Ke.

dS (a, 2) dS (B, %) dS’ ae dS (y,%)

and suppose ¢=/, a +h: gar +f/s —— + &e.

Now when «>a dS (a,x) _, dS(B,2) _, dS (y,%) _,) 9
eo fl then = al =! Meda Cait Teeny =0, «ec.

and, by hypothesis, ¢ is then=4q,.

3C2

382 Mr Mourpny on the Inverse Method of

Again when => dS (a,x) _, dS(,3) aS y,2) _ :
ee then gee =] SS =i dy =0, Ke.
and @ is then=¢,;
and so on; if we substitute these values in the assumed for-
mula for ¢, we get successively the Equations,

P=h | and therefore ji=¢:
pa=fhtf ! h=bh— oi
pa=fht ft hf | A=
&e. =e. | &e. = &e.
and. p=, SI) + (9,—4,) an + (ps— ds) Se +8 &e.

Cor. If the function is also discontinuous below a, as *
passes through the decreasing magnitudes a, b, c, &c. ¢ assuming
then the values ¢, $-1 $-2, &c. it is easy to see that then

dS (a,
e=(o-9) SI? +(g.-g) SE

1. x %
|+(.-9-) =e + (p_1— p-2) db + &e.

(3) Geometrical Illustrations of the theory of Discontinuity.
(Vide PI, 24).

23. It only remains to add a few examples, to illustrate the
application of the principles of this Section.

(Fig. 1.) Let 4Ca, BCb, be the equal sides, produced, of an
isosceles right-angled triangle; we may thus find the Equation
of the black part ACB, as distinguished from that of the dotted
part aCb,

Definite Integrals, with Physical Applications. 383

Make the middle point 0 of the hypothenuse, (the length of
which we may suppose=2,) the origin of co-ordinates; the axis
of « being the hypothenuse itself.

The Equation of 4Ca, generally, is......y=1+a
Le sciees cole eae s espe OUP CDs. ccs es.0e0 IS 22.0.4 = 1 — a,
and therefore the general equation of the system is
(y—a—1) (y+a—1)=0.

in which, any value of x, as OP, corresponds to two values
of y as PQ, Pq; the least of these two values is that which
belongs to the black part ACB, at both sides of the origin:
the equation peculiar to this part is therefore y =the least of

ee substitute in Art. (16). 1—# for a, and 1+2a for 6, and we

get for the required equation
1.1

eh rar . (1— 2°)’ +

Led! 9

aaa) + &e.

Ets)

We may now treat this value of y, by the ordinary methods
of analysis, for the attainment of any object which has _ sole
reference to the division ACB of the given system.

Thus to find the area of the triangle ACB, we must take
as usual /.(y) from «=—1 to z=+1.

Now the general term in the value of y is

WA ai. fe

DIAM OF Ssece secs Qn (T= 2")

the integral of which from «= -1 to x= +1 is
g 1 1

(@n—1).n41) °°" Q9n—1 2n+l’

384 Mr Morpny on the Inverse Method of

put 1, 2, 3 &c. successively for x in this expression, and we shall
generate the series for /.(y), namely

1 1 1
rtgtgt7t ke.
istha sa abe
—3-3 7 7t &e.

a result which may easily be verified, by geometrical considerations.

24. We shall terminate this Section with an example, in
which the function is, infinitely, discontinuous.

Fig. (II). An infinite series of equal and similar arcs of any
kind, are ranged at equal distances, across the horizontal right
line Aa, the successive ares being included between pairs of
equidistant ordinates; to find the equation of the system.

(Note, the lower branch of Fig. II, only differs from the upper
in supposing the extreme points (a, 4:) (a, 4.) &e. to coincide).

Let y=/(«) be the equation of the curve of which q@c,d is
an are, « being the abscissa for the curve when continuous, and a
the abscissa for the discontinuous curve; let the distance c,c,=1.

When a is between - § and 3; y=(f(a) is the equation to a,c,
Sicues assed ebaeegcnseage cab. Paneas) YA fla— Besss.. cases. es s..0¢e) 8,

sotteeBetcbeas stevaipena <scagne Spd ee Me alate BK, 5, Fo sc ouch ode vdnee On CyDs.

Similarly,

when « is between — } and —33 y= f(at1).....ccccceeseeseneeeens@1 C1 0-1

&e. &e. &e.

Definite Integrals, with Physical Applications. 385

Hence, if L be the least of the quantities a, a+1, «+2 at
a—1, a—2 &e.
then y= /f(L) is the required equation.

Now ZL is evidently = the least root of the equation sin x (w—a)=0.

Hence as in Art. (18) we have

sin 7 (#—a)

a Lf (x) |

aim

where ,f’ (x)

y = coeflicient of + in—f’ (x) h. 1.

or if we put

F' (0) (pO)

ar E=9 _2= g(x) then y=/(0)—S (0). (0) +{2 -0|

a COs (a7)

* Fourier has expressed discontinuous functions by means of
Definite Integrals; and when it is unnecessary that their form
should be explicit, his results may be conveniently used.

- * Vid. Fourier, Theorie de la Chaleur.

386 Mr Morpny on the Inverse Method of

SECTION III.

APPLICATION OF THE PRECEDING PRINCIPLES, TO THE PHENOMENA
OF DEVELOPED ELEcTRIcITY. (Plate 24).

25. WueEN a body is electrised, either by communication, by
influence, or in any other manner, from known or unknown
causes, the external action is subject to observation. The law
of the action of an electrical particle, at different distances, is
at the same time, known to vary as the inverse square of the
distance. To find the nature of the electric distribution necessary
for the production of any given phznomenon, is evidently a
mathematical problem, the solution of which may easily be sub-
jected to the test of experiment. As it is also peculiarly adapted
to illustrate the analytical principles which precede, we have taken
it as the subject of this Section.

26. (Fig. 3.) For the external action, it will be generally
convenient to substitute the electrical tension at the different
points (Q) of an infinitely thin, conducting rod (Bz), communi-
cating with the electrised body APB. This tension is the sum
of the quotients when we divide the mass of each electrical
particle, by the distance of that particle from Q.

27. Suppose the figure 4PB spherical, and that the electric
tension in the rod Aa passing through the centre, at any point
Q in the external part is found to be inversely as the distance

Definite Integrals, with Physical Applications. 387

O® trom the centre of the sphere; and the law of electric accu-
mulation at the surface of the sphere is required.

Make AB the axis of x, the extremity 4 of the diameter
being the origm; the element of the surface included between
two planes at the indefinitely small distance 52, and _perpendi-
cular to 4B is then=2ra.dx, a being the radius of the sphere;
and the distance of any point in this annulus from Q

= fa? +B?4+28.(a—2)}*
putting OQ=~, and if we represent the electric accumulation by
A, which is in this case a function of x, the expression for the
tension on any point Q of the rod AB is

QraA
J aap. Gray from #=0 to v= 20,

neglecting the indefinitely small action due to the electricity on
the infinitely small surface of the rod.

5 A
ie Secs rc SET EE

and therefore by the proposed condition we have

m _ A
B 7 Awana oes: (i—20}8”
when a<f, or (Gaara Sai :
1+ a2 +5

@
p
Vol. 1V. Part III. 3D

equal a constant quantity m, = being any proper fraction; from

388 Mr Murpny on the Inverse Method of

which we get m= j|,(A) (putting 3 =)» as also that for internal

points, where 2 is a proper fraction

“_f 2
a “4 fa+P'+2aB.(1—22)} ’
from which premised observations, the question takes this form:

To find (A), such a function of ¢, that we may have

=e ;=the least of \s

lee PP aaR AS 3H]

B
The left-hand member of this equation

when expanded becomes
9? ap 1 arene 2! a? B
ix
idl agtt @ arap !t 2a: GBP +&e
The right-hand member expanded by Section II, Art. 17,
becomes

1 1.1 Vas lo a3 ‘
nla + 2.4 (+p) * 2.4.6 (ath) &e-}

and comparing the general terms in both series, we have
3...(2¢—1) 1.1.3...(2%—-1)

At. — 5.2, ee

f eae = Saag (deta). >
m

or fA.e= Pay

and therefore by the principles of Section I, 4=the constant
quantity m.

Definite Integrals, with Physical Applications. 389

In fact, when the electric accumulation is constant the ex-

ternal attraction will evidently be =~; and the integral with

respect to 8 which represents the tension of the fluid at Q is

evidently then = : :

28. The law of accumulation in the case considered above
is the simplest possible, but when the sphere is subjected to the
electric influence of other bodies, the function which expresses
that law may have any form (subject to vanishing at infinite
distances), the indication of which form is to be had in the law
of the attraction on external points, or its integral, the tension
of the electricity in the infinitely thin rod ABz.

To generalize the preceding investigation we shall take the
following example, which leads to some curious results.

The tension in the external rod, varying as any inverse power
of the distance from the centre of the sphere; to find the law
of electric accumulation on the spherical surface. (Fig. 3.)

Adopting the same notation as in the last article, and repre-
senting the law of external tension by

” fF a” A
m. gar Ara’, we have when a<[ mo = (eS

and therefore for internal points where a>, we may put 8 in

. . . a mm @
this expression for *; and thus we find the law of internal

tension to be m. Wild 4a’: that is, when the external tension
a

+1 *

8$D2

390 Mr Morpny on the Inverse Method of
o me the internal tension <"; the proposed question now be-
comes

To find (A) such a function of ¢ that the value of

(ee may be m Lae ee
fa +B +2aB.(1—2aye? AY Be oe?

according as a is less or greater than #.

The formula which represents the latter discontinuous function
(by Section II, Art. 20.) is a series of which the general term is

1.3.5...(2@-1) — w.(w—1).(a—2)...(v—n +1)
2.4.6...2% ~ (+1). (v+2).(v+3)...(a+n+1)

g%+1_ (aB)"

m. Son
(a+6)?"*" F)

. (Q@n+1).

and the general term of the integral when we expand the deno-
minator is as before

1.3.5...(2%—1) 2?*.(aB)*

hAt. ~O76...@a) (at Byer?

and if 4 receive a yalue such as to make the general terms
of both series identical, the series will also be themselves iden-
tical.

Equating we get

f{Aea=m(2n+1). (a +1).(@+2).(@+3)...(a++1) ’

from which we may find 4, by the method given in Art. 8,
Section V. (Vid. Ex. 4.); its value is

é n n+l n.(m—1) (n+1).(n+2) ,,
A=m(2n+1).(—1).j1- 7: i ae er mE ae ahaa .C—&e.}

Definite Integrals, with Physical Applications. 391

29. If we make x=1, then the law of internal tension is ex-
pressed by 47m, and therefore the internal attraction is the con-
stant quantity 47m. Hence if a sphere AE BF (Fig. 4) be electrised
by the influence of a very distant body P, the force exercised by
P to separate the combined electricity in the interior of the sphere
may be represented by the constant —47m. The law of accu-
mulation is then 4=3m(2t—1) which vanishes when ¢=4; there
is therefore a transition line EF, which is a great circle having
its plane perpendicular to the direction of P’s action; this line
divides the sphere into two parts containing equal quantities of
the opposite electricities; the maximum accumulation takes place
at A and B the poles of the transition circle.

30. The expression for the accumulation 4 in the general
case of Art. 28, may be put under an elegant aud simple form,
which facilitates much the ¢racing of the actual distribution.
If we observe that

ad?" i d”,¢e+?
1= 9 3..ndp and (+1).t=T 9 gp &e- we get
_ m(2n+1)(—1).d , Sr ee eg
Aa mate. nde ia hE nave iainit &e.}

_ m(Qn+1)(—1)" d"
a et te

(¢¢)” puttmg f=1-¢.

31. To determine the number of transition lines on the sur-
face of the sphere put

Pee d' 4 eee a ; r 3
reo ae) =P,, then 4=m(2n+1)(—1)'.P,;

and therefore the equation for determining the transition lines
is P,=0.

392 Mr Morpny on the Inverse Method of
Now it is evident that

1.2.3.. n-1).Pr= 7

{(¢¢’)"-?.(1— 22)}

d- d"-*

=(1- 22). 7 o~ (tty 2.(n—1). Fame (tty

P,=(1-2%).P,_,—2(n—1). f Py.

Suppose now that for a particular value of x, all the roots
of the equation P,-,=0 are real (as 4, &, ty.--¢,.:); they lie be-
tween the roots of the equation {P,-.=0, and from the nature
of the question it is obvious that they all lie between 0 and 1;
substitute them in the order of their magnitude for ¢ in the above
equation, then since P,-, vanishes, the resulting values of P, will
have alternately opposite signs. Again, when ¢=0, P,=1=P,-,;
and therefore P,_,, and consequently its integral remains positive
from ¢=0 to ¢=4, the least root: the order of the signs of {P,_, is
therefore +, —, +, —5--¥-0. the last in the series bemg + or —
according as ” is even or odd, and the corresponding signs of
P, are —, +, —, + &c. in number (x—1) also. If we put ¢=0,
P, is then =1, and therefore its sign is +; and if we put ¢=1
then P,=(-— 1)"; it follows therefore that the substitution of
0, 4, tytn 1, m P, for ¢ gives 2 alternations of signs, therefore
the roots of the equation P,=0 are all real on the supposition
that those of P,-,=0 are such. .

Now by Art. 29, there is one transition line when »=1, hence
there are two such lines when x=2, 3 when x=3 &c. so that
generally there are x such lines symmetrically situated with re-
spect to the points 4 and B, (vid. Fig. 5, where the black lines
represent the transition circles, which divide the surface of the

Definite Integrals, with Physical Applications. 393

sphere into x+1 portions, containing alternately the positive and
negative electricities).

32. It is obvious that in the x+1 portions into which the
spherical surface is thus divided, there will be x—1 lines in
which the accumulation of the respective electricities is a maxi-
mum, beside the poimts 4, B; (represented by the dotted lines
in Fig. 5), they possess the remarkable property, that with the
transition lines, they divide the surface into belts containing
alternately exactly equal quantities of the two electricities. Thus
the electricities in the portions LAF, E,F'\b,a, are equal and of
opposite kinds, the electricities in @F,E., and a.b.F.E. are
similarly equal and opposite and so on.

For the whole quantity of electricity, from the point 4 where
t=0, to any value of ¢as 4

= fina. A=Arma (Qn +1)(—1) f P..

Now by the theorem of Leibnitz
d’ (uv) dv dv d'~’ wv

Faas ar age
(phd sia 2b
we get LPn— Ton de>
bs (n—1) ” jn=1,, -(m—1).(n—2) n.(m—1) n-2,,
the integral being supposed to commence from ¢=0.
ae dP, 1 a(t? M+1 NM ,n-1
Similarly di id,..n edpmersis made bie
(n+1).m m.(m—1) »-2, (a+1).m.(m—1) n(m—1).(n—2) ,x-5 5 0
eae, eer eee tae Fe:
tt dP,

and comparing both, we get f{ P,=— aad) di

394 Mr Murpny on the Inverse Method of

Now for all the maximum values of P, we have GT ~ 0,

hence {P, vanishes from ¢=0 to ¢= any value which makes P,
a maximum, and also between any two such values, from which
it is evident that the positive and negative parts of those in-
tegrals are exactly equal.

33. Let us now suppose any law % Samy tension (vanish-

B.a

ing at infinite distances) as f+oe+5 + &e. and seek the law

ao
of electric accumulation, necessary to produce it.

It is evident from what has preceded, that the corresponding
ai BB, C. a

law of internal tension will be

Now the

laws of accumulation necessary to pets he tensions

(4 and 4), (42 and 2P), (= and £5 &e.

and if we suppose the electrical systems which correspond to
these laws of accumulation to be superposed into one system,
and the accumulation at each point estimated by the excess of
the positive sum above the sum of the negative electricity at
that point, it will be the arrangement which was required to
produce the given phaznomenon.

The accumulation therefore is

ditty: 5C @itty’
{A— 3B. ar Fie t310° 9F . —&e.},

and therefore at similar points in different spheres it is inversely
as the surface of the sphere.

Definite Integrals, with Physical Applications. 395
34. Let this law of accumulation be expressed by the form

ing While the law of external tension is aw (5) ;

Now ¢=4-3B. ater) + BO Gisney

dt 12° ae ~&:

a’

=term independent of 8 in {4 +B.G +C. mt &e.}

B ditt’), B a(t &e.}

ft 5
eal dt @'1.2.dé

2 |
“

d (tt') (ee CP a
Sy een ea a

i SRT
=coeflicient of AM; vel
aaa

And if we form the equation w=¢-— Bum), we get by the

theorem of Lagrange

d
2 =, (tt)
CEE: thagk di
tse — iz Mb) taining em
d
£ (ity
: d Peres Bo pi dt
a) ap hw) =48 y— F — (tt) 45.75. 1.2

and .*. ¢=coeflicient of 5 in {5 r(5)} 5 eS {gps ay (Biwi

: gaa h ae a\' d du
=coeflicient of BM a a ‘ap (a. =) ;
Vol. IV. Part III. 8E

396 Mr Murpnuy on the Inverse Method of

But the least root of the equation
B

u=t——.u.(1—x) is
a
Bta 1

23 2B

“= {BP +a? +2aB.(1—2A)t2,

du a :
and therefore dt > if +o'+ 208.1 2aft?

d A os aa age paar te
and 7g (8! Ge) = apt TB Fas Tap 8H

z= : Ws a a — 3°
Hence ¢=coeflicient of B in B SG) eC aT

If therefore we multiply the external tension by

a? — 3?
{B+a°+2afB.(1—2¢)}3

and divide by 47a the coeflicient of i in the product, the result

expresses the law of accumulation.

Or if we multiply the internal tension by the same factor
and divide by = the coefficient of . in the product, the result
will likewise give the law of accumulation.

This may be expressed in a definite integral by the theorem
(Camb. Trans. Vol. 111, p. 438.)

35. When the electro-motive causes are unknown, then the
application of the first of these rules to the observed law of
external attraction, or tension (which is the integral of the at-
traction) will determine the law of distribution. But when the

Definite Integrals, with Physical Applications. 397

above causes are known, if represent the electro-motive force
at a distance 8 from the centre, then—/;,f must be the internal
tension produced by the action of the sphere, and the application
of the second rule will then give the law of accumulation.
We must however always put the internal tension under the

form 2 v2 (2) j

Thus in Fig. 4. if the point P separates by influence, the
combined electricities of the sphere, and Q be a point within at
a distance 8 from the centre, then if we put OP=ha the force

on Oo aaa to destroy which the force exerted by the elec-

;> and therefore in this case

aes
(B—ha)

tricity on the surface must =

and therefore the law of accumulation is

cali ee Ro 5 d?.(tt’)
a li- dt *1.2.8° dé -&e|
ch d 1 1 d , 1 a? ; :
~ Ora an iim Te ae + ae: ap &e.}
cht d 1 du 1
moO ah (a a) , when w=t— a )
7 chi ad fr ase
aio? <a VAs {14+2h.(1—22) +h'33

c 1-h?
Ana’ $142h.(1- 24) +A7t8°
8E2

398 Mr Morpnuy on the Inverse Method of

Using the notation of the last Article, it is evident that the
whole quantity of electricity in the sphere, i. e. the difference
between the quantities of positive and negative fluids, generally,
is equal to /{i:¢ = coefficient of

1A es [fC a’ — (3° - oe ee
B in Bl ee trom ¢=0 to ¢=1t.

Now this integral = hay ble ->. Eu

ce “a8 la=B. Bra
and therefore the quantity of electricity = coefficient of F in

: S At 3 :
the external tension, or of ; im the internal; thus in the example
tL

before us this quantity = — 7 but if it were necessary that there

should be a given exeess EL of positive above negative electricity
on the surface, we have only to superpose the uniform accu-

E+5
mulation qa te the variable accumulation expressed by
a)

Bard. (1—h’)
Ama?” $1+2h(1—2th+ htt?

To find the position of the transition line we must find ¢
in the equation

Sab h?—1
er ae {14+ 2h.(1=20) + h?}3°

When the electricity is excited merely by influence, we must
put H=0, the law of accumulation is then as

1 h?—1
ho 314+2h.(1—2t) +h*)2”

Definite Integrals, with Physical Applications. 399
that is, the accumulation in any annulus E’F" is proportional of

the constant PEPO-PH above the variable ——— Ta

36. In general whatever be the surface of revolution, if we
put the length of the axis = 2a, and x=2at, y will be a given
function of ¢, as well as the annular element of the surface

4.
2 rN SS
Ty J/1+ 0 .o¢, or S.d¢; the tension in the axis= Ura TGlane

and expanding both sides in similar forms and equating we shall
have {AT.P,.t¢=P,, P.t’ and P, being the general term of
both expansions;
-. by Section I, 4= se eacticient of 1 in ee
1 x P,

The known properties of Laplace’s coefficients might have
been employed in deducing some of the results given in this
Section; but the principal object was to illustrate, in an inter-
esting subject, the application of the Inverse Calculus of Definite
integrals.

400 Mr Murpuy on the Inverse Method of

NOTES.

Note A—(Vid. Art. 9.)

(1.) Application of the Inverse Method to Logarithmic Functions
m general.

To complete what was observed in Art. 9, let P and Q denote the
same quantities as in that Article, and let R be a function of «x which
remains finite when @ is infinite, i.e. of the form

Mee ey he:
ie

we are now to find f(¢) when ¢(#)=f#.h.1. a :
Now by the general theorem (Art. 5.) we have the Equation
: a a P
tf (t)=coefficient of 3 in R.t hl. oo a (1.)

Suppose #.é-" to be actually arranged according to the powers of 4,
it will consist of two parts, one containing the positive powers of 2 and
the absolute term, all which we shall represent by S'; the other, contain-
ing the negative powers of a, suppose S’; hence Equation (1) becomes

mene tf (t) = coefficient of 2 in (S'+8’) h.1. =

but since h.1. i when expanded contains only negative powers of 2 like

S’, we may reject the term S” h.1. z= altogether, therefore we need es-

Q
timate the value of S' only, and we shall thence determine

J (t) by the Equation ¢/(¢) = coefficient of = in Sh. 1. Qo @)

Definite Integrals, with Physical Applications. 401

ing value of /(¢) be put = 7 when determined by the methods given
in Section (1).
Hence 7'=coefficient of in aie
=term independent of x in Ré-*;
oo Jf = =term independent of 2 in a
= coefficient of «in Ret“.

Similarly f ; f = =coeflicient of a in Rt*
t t

~ {3 fs . [ F=coetticient of «? in Re.
te é¢ t
&e. = &e.

Having thus obtained the absolute term and the coefficients of x, x*, «°, Xe.
in Rt", we have therefore,

Poli hal schilintie Vogl ga
S=T—x [> +a Sele? Le LG LG t+ 8)

all the integrals being made to vanish when ¢=1, for then Rf‘= Rh,
which contains no coefficient of x, 2°, 2°, &.; the value of Z' is then
= term independent of x in #; that is, A.

By differentiating the Equation (3) and substituting S for its value

we get
aS: «dT 2S Minis) sys oh dT
Fee Poe er ats reget eer df’
dT

and integrating we have S¢*={¢ aa? the integral being corrected by

the condition S=A when ¢=1; the value of S' is therefore fully discovered.

402 Mr Murpny on the Inverse Method of

Recurring now to Equation (2), let {.S' be represented by F'(—2)
so that F’ denoting the derived function of F', we have S= — F" (-2);

-. tf (t)= — coefficient of + in F(-2).W1G

= —coefficient of i in F'(—2). {hu a —h.1. a,
2 a x

and, by a property of equations demonstrated in my former paper in
this Volume (p. 138.), this gives us

Sf (t= 4 LE (a) + F(a) +o. + FG) =F (by) - Fb) = Fb}.

If we substitute differential coefficients for integrals, a process similar to
that by which S' was found, will determine S$’; the value is given by the
‘ ; LT : : ad
equation S’¢? = — ft TF the integral being corrected by the condition,
that S’=R—A when t=1, and we may verify the results by adding the
values of S' and S” which will give $+ S’=R.¢™.

(2.) On the separation of the positive from the negative powers,
of the variable; in functions generally.

In the preceding example, the form of the function R¢" contributed
materially to facilitate the calculation of the parts which involved the
positive and negative powers of « respectively. As on many occasions
it would be useful to effect this object, we may here insert a general
method founded on very simple principles. i

First, let us seek the term independent of #, in a function of 4
containing both positive and negative powers of #, but composed of a
finite number of terms.

Represent by F’'(h) the given function, and let be a number greater
than any exponent of h, positive or negative which it contains.

Let the z™ roots of unity be

27 2.(n—1).7 —
Vs) Qh, Gejeveon-dnay OF 1,.€% Che t

eeenee

47
N=1 ex V-1

Definite Integrals, with Physical Applications. 403
and suppose #’(h) when arranged according to the powers of h to be
=4A,+ Ajh+ Ah’ +...+ A_,.h-'+ A_sh-* + &e.

it remains to determine 4,. Putting for the above roots successively
we get
F(1)=4,+ A+ 42.+...+A_1+ A_.+ &e.

F(a) =A, + Aya, + Aza’? +... + a + ao + &.
1 1

F'(a;) =A, + Ava: + Ayas + fo Teen a + &e.
&ec. = &e.
adding all these equations and observing that
1+a,+a,+ &€.=0

l+a,+a,+ &.=0,

we get 4,= ee

Secondly, when the number of terms in F'(f) is infinite, we must put
a=, and observing that generally

ett) anf —1

it BE pay pBavsa) ee (2
n n

SAL
4nrV—]
+ F(e a), 227V=1 5 ge,
n

Vol. 1V. Part III. 3F

404 Mr Morpny on the Inverse Method of

it follows that when 2 is infinite
1
A, = ——_—— {, F'(c*) from 0=0 to 0=27rv7 —1,
a oe vig
which accords with the result given in the memoir on the resolution of

equations, Section V. in this Volume.

Lastly, we can now determine in any function of w as f(«), that part
which contains only positive powers of x, and the absolute term; thus let

J (e)=B,+Bi.0+ Biv’ +... + B_,.027'+ B_..2-* + &e.

ae he
ee (7) _ {B+ B.5 + Bus tae +Bos75 5

x{l+h+h'+ &e.},

and selecting the parts of this product which are independent of h, we get

B+ Bix + Bex’ + &.=term independent of h in i

—

Vein = 0 to 0=2rV/ — 1.

Note B.

(1.) On the apparently improper Forms of > (x)

Ir has been shewn in the first Section, that when /(¢) is any of the
functions usually received in analysis, ~(a) must converge to 0 as 2 ap-
proaches «, and therefore consists of essentially negative powers of 2; it
often happens however in the application, that (2) presents itself under
apparently a different form. Let us examine the difficulty thus offered;
and the simplest mode of doing this, is to take a specific example.

a a zl
pop tes Fete ravh.l. 4%

Let $(2)= y + to find f(t).

Definite Integrals, with Physical Applications. 405

The last terms in (x) seem to be of an improper form as involving
positive powers of a, the first terms are also such that (2) apparently
becomes infinite for values of 2 between 0 and o. All this however is

ee its value

merely in appearance, for if we put for h.1.

1 1 i 1 1

@ 8a 8e@ 7 "PP eae, (@+1).a"** (@+2).a? wi

we see that the real value of ¢() is

A still more striking instance of an apparently improper form of p (2)
is the following:

Given (#)=1-a+a.(a- 1)—a.(a—1).(a—2)

+...4+4.(@—-1).(2- 2).3.24Fa@.(e—1).(a—2)...9.1, (1 - *).

This function appears, at first sight, to consist of only positive powers
of «; while in reality the function consists strictly of negative powers

its value,
1 1 Lean: 1
aS} ilps Wa Borat at T.8,..e4d)

and we get (x) in the proper form; viz.

Pang se 1 1 it
(Acdsee 1 (a +1).(~ +2) a (w +1).(% +2).(@+3)

3F2

&e.

406 Mr Morpnuy on the Inverse Method of

and therefore by Art. 8. Ex. 5.

tah ae

f()=1— (1-8) + SA = 5 thee

Consider lastly the case of p(x) =(a+x)t—(a«—a)}, to find f(t).

3
Observing that (#—a),=(a+2)'. (1 — a and expanding, we have

1.3 a
2.3 pas ele

$(*)= Gra t 1.2° @saye

a 1.1 a ne
1.2.3

and therefore by Art. 12. Section I.

a1 f@(h. 1. t)74 1d @ .(h. 1. £)4 1.3 a’.(h.1. é)3
Paige ae +e (Lae +735: fib. La we.}
#- (Qa(h.Lé)-y (2ay?.(h. Ld)! (2a)3.(h.1. 23
ee a sanaiesan —fe.}

fo) ee

= at at
1 t*—t%
= Ont eho

(2.) Method of estimating the Values of Operative Functions.

Ir any function of a be reduced to the form of a sum, composed:
of terms which are other functions of 2; the identity thus formed will
continue to subsist, when a is changed into 2+ in both its members.
Let the operation which indicates that 2 is to be changed to +h be
denoted by ¥; let "denote that 2 must be changed to a+nh and
vy", into #+ = then y" is evidently equivalent to the repetition » terms
of the operation ¥, while the latter is equivalent to the repetition x

1 .
terms of W"; if beside the operation y" the function of « is also to be

Definite Integrals, with Physical Applications. 407

multiplied by a constant quantity 4,, we may represent the combined
operation by 4,\", and the sum of any number of such operations as

A, W" + A," + A, + &e.
may be represented by the operative function F(y).

Suppose also that p(#) the given function of w which is subjected
to the above operations, is by the inverse calculus reduced to the form
kf (t).@3 the result of the operation F’(y,) will be the sum of the
partial results of the same operation on each element /(¢).¢* . d(¢);

but v7 (é*) =e" = (¢")". (é*) ;
edt (ab \ote at (2). 5
fs F().¢(2)= 1 Vibe! ad Can ar eae ee (1.)

Tet the operation which is the inverse of F'(\) be represented
by F’-'() so that the latter operation performed with the former neutral-
izes it, then

Ue ea Cet = Cats (a). 8s
let the roots of the

EY (e"\=0 BE 24, Mtg, Magoo 000+. 8Cn 5

w Fb) {.4, ™* + Ape" + Aze”s” &e.} =0, where A,, 4., A; &c.

are arbitrary constants ;
oo F (Wy) (0) = A, 6” + Ao 6" + &e.5

which appendage must be added to the operation F'(\) ¢ (2), since # (2)
is analytically to be treated as (a) +0.

It is evident that the operation y/’ is equivalent to multiplying the
given function by unity, and therefore ¥/-—y is equivalent to the opera-
tion A of finite differences; consequently y is equivalent to A+1; Hence

ECA). (2) =f BP = 1) f(b). brerecnsepnece(B)5

408 Mr Murpny on the Inverse Method of Definite Integrals.

writing for A in this equation, and making / indefinitely small, when

Awe d an |
738 usually represented by qe and oe =h.1. (¢) we get
F(4) ¢(e)=ff).F (hl. (3)
dz)” if (é). Ra Ae oa POE a F

where the appendage is, as before, to be annexed.

To reduce functions containing positive power of «x, and zero to the
form /f,f(¢).¢° belongs to the next part; but we may here indicate, that
the series given for discontinuous functions, may be used in this case,
to reduce the proposed functions to forms to which the principles of this
part may be applied.

It was necessary to make the preceding remarks on Operative Func-
tions, to explain more fully the allusions in reference to this subject in
the Introduction.

Caius COLLEGE, R. MURPHY.

May 26, 1832.

XIV. On the Phenomena of Newton's Rings when
formed between two transparent Substances of dif-
Jerent refractive Powers.

By G. B. AIRY, M.A. F.R.AS. F.GS.

LATE FELLOW OF TRINITY COLLEGE, AND PLUMIAN PROFESSOR OF ASTRONOMY AND

EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE.

[Read March 19, 1832.]

In a paper communicated to this Society about four months
since, I stated my expectation (founded on Fresnel’s theory) that
if a lens of a low-refracting substance were placed on a_ plane
surface of a high-refracting substance, and if light polarized in
the plane perpendicular to the plane of reflection were incident
upon it, then so long as the angle of incidence was less than
the polarizing angle of the low-refracting substance, or greater
than that of the high-refracting substance, Newton’s rings would
be seen with a black center; but if the angle of incidence was
greater than the first of these and Jess than the second, Newton’s
rings would be seen with a bright center. I have now to an- _
nounce the fulfilment of this anticipation.

Before describing the method by which I have succeeded in
the examination of these phanomena, I think it right to give

410 Proressor Airy on the Phenomena

a theoretical calculation of the intensity of light in the rings:
as without this the necessity for some of the precautions will not
be sufficiently evident.

Conceive two nearly parallel plates of different media to be
separated by a plate of air whose thickness is 7; and let the
vibration in the plane of reflexion, of an incident stream of

light within the first medium, be represented by a sin (vt — 2)

where « is the equivalent in air to the actual distance of a par-
ticle from some fixed point, (the light being supposed polarized
in a plane perpendicular to the plane of reflexion). Let « be the
angle of incidence on the last surface of the first medium; ’ the
angle of refraction, which is the same as the angle of incidence
on the first surface of the second medium; and «’ the angle of
refraction in the second medium. A part of the light will be
reflected at the last surface of the first medium: a part will reach
the first surface of the second medium, where it will be sub-
divided; and one portion will be reflected to the surface of the
first medium where it will be again divided, and one of its parts
will enter in the same direction as that which was reflected at
first. In this the phase of the undulation will be behind that which
was first reflected by the quantity corresponding to the space

2T cos’: or if = (vt—-2) be still taken as the measure of the phase

2 4 :
of the ray first reflected, = (vi—2) == Tcos:’ will be that of the
ray which has been reflected at the surface of the second medium
and then enters the first. The quantity = T eos: we shall for

abbreviation call VY. Of the light which reaches the surface of
the first medium, a part will be partially reflected at the surface

of Newton’s Rings. 411

of the second medium, and will partially enter the first medium:

: ; 2 : :
its phase will be FZ (eé-2)-273 and so for succeeding reflexions.

Now suppose that at the last surface of the first medium, the
coeflicient of the incident vibration being 1, that of the reflected
vibration is e and that of the refracted f; at the first surface of
the second medium, suppose the coefficient of the reflected vi-
bration to be &; and for light incident from air on the surface of
the first medium, suppose the coefficients of the reflected and
refracted vibrations to be hk and &. Then, the coefficient in the
incident light being a,

That in the first reflected light is........ ae
that in the refracted light is af
that in the light reflected at the second medium is afg
and that in the light refracted into the first medium is........ afgh

that in the light reflected from the second medium is afg°h
and that in the light refracted into the first medium is........ afg"hk
and so on; the coefticients after the first following a geometrical
progression whose ratio is gh. Thus it appears that the whole
vibration will be

qe _ (2a - (27
@.e.sin =" (vt-2)+a.feh {sin (= (vt—2)-V) +gh.sin (= (v-«)-27) +&e.l,

sin (= (we-a)— v) —gh.sin (= wea)

.- Qa
ol a.e.sin—(vt—a) +a.fgk. I—agh.ca Vig

Now in Fresnel’s expressions,
_ tan(«— v)
~ tan(+2)’
Vol. lV. Part III. 8G

412 Proressor Airy on the Phenomena

_ cose (/, tan («—v’)
Nig cos’ ( tan a) :

_ tan (’—0”)
~ tan (/ +1)”

_ tan (i —2)
~ tan (+2) ?

cost’ ( tan =a)
k= - ———}.
COS t tan (c' +2)

Hence fk=1-—e, and gh=—ge; and the expression becomes

we sin (= (wea) - v) +ge.sin (= (vt—2))
ry oe ae: 2:
awe sin (vt x) +ag(1—e’). Tape ue Ph ae

Resolving this into the form
Psin = (vt—2) +Q cos" (vt—2),

the intensity or P’+Q’ becomes

eo tet+2ee cos V
1+2gecosV + g"e"

The maxima and minima of this-correspond to the maxima
and minima, or the contrary, of cosV?. When V=0, 27, &c. that

5 r 2r 5 § :
is when 7'=0, or = RoR OF = aco &e. the intensity of the

reflected light is

deere

of Newton’s Rings. 413

r 3X
4cos.’ 4 cose

“(Ee

1-ge

and when 7'= » &c. the intensity is

and the excess of the latter above the former is

Se eee) = 2")
he pissy au

This is the difference of intensity of the brightest and of the
darkest parts of the rings: and when it is positive, the center
of the rings is dark.

Now tan’(.+¢) is always greater than tan?(«.—/), and tan’ (: +c’)
is always greater than tan’ (—¢’): so that (l—e’).(1-g*) is always
positive. Consequently the central spot is black when e and g
have different signs, and bright when they have the same sign.
Or as tan(:-/) is always negative and tan (’-:’) always positive,
the central spot is black when tan (+) and tan(’ +c’) have the
same sign, and bright when they have different signs: that is,
it is dark when «+ and ¢ +’ are both less or both greater
than 90°, and bright when «+/¢ is less than 90° and ‘+’ greater
than 90° (or vice versa). From this it follows that while the
angle of incidence is less than the polarizing angle of the first
medium, the central spot is black: at that polarizing angle the
rings disappear (as e=0): from that angle to the polarizing angle
of the second medium the central spot is bright: at the polarizing
angle of the second medium the rings disappear (as g=0); and
beyond that, the central spot is again dark.

Now let us estimate the intensity of the light at the central
spot when the first rig is black (the angle of incidence being
between the two angles of polarization). If the first ring is black

862

414 Proressor Airy on the Phenomena

g-e
1—ge

spot becomes a’. (-*%.). The condition g=e gives
P 1+é

we have =0, whence g=e: and the intensity in the central

tan (c’—v’) _ tan («—0’)
tan +c") tan («+V)’

whence sin’ 2’ =sin 2. sinc”:

1
or cos?’ = —— cosr.cose”.
mm

where m and m’ are the refractive indices of the two media.
Without attempting to solve this equation generally, suppose
m=1,53 and m'=2,45 (which correspond nearly to plate glass and
diamond). The values of ¢ at the polarizing angles are 56°. 49’. 54”
and 67°. 47.48"; and the value of ¢ which makes the first ring
black is 63°. 19’. 4”; the values of « and «” corresponding to this are
35°. 43'.57” and 21°. 23'.21”: whence e=g=0,083215; and the in-
tensity of the light at the central spot=a’ x 0,02732.

But to obtain a practical idea of the import of this expression
we must compare it with the intensity of light in the rings
in some other position. Now when the incidence is perpendicular,
the expressions above give for the difference of the light in the
dark spot and bright rings, a’ x 0,28159. Consequently the inten-
sity of light in the rings seen between the two polarizing angles
is less than one-tenth of that in the rings seen at a nearly per-
pendicular incidence. As the latter are by no means vivid, we
must expect the former to be faint.

The intensity of the rmgs which would be produced at the same
angle of incidence by light polarized in the plane of reflection,
sin (c — =)

sin (c’ + hh

sin (:—”’)

found in the same way, (putting Fim Gane) #7)

and g’=

of Newton’s Rings. 415

is @ x 0,66487; and is consequently about twenty-four times
greater than that of the rings of which we are treating.

This shews that much care will be necessary to make the
rings visible. Suppose for instance that the incident light is po-
larized by a plate of tourmaline, or (which amounts to the same
thing) that the reflected light is examined by a tourmaline, with
its axis perpendicular to the plane of reflection. Few tourmalines
are so perfect as to transmit no more than one twenty-fourth
part of the light polarized perpendicular to their axis. If then
the rings are examined with one of these, the rings of which
we are in quest (whose center is bright) will be mixed with rings
produced by light polarized in the plane of reflexion (whose
center is black) of at least equal intensity: and their character
will therefore be entirely destroyed. If instead of a tourmaline
we use a doubly-refracting prism, with which both sets of rings
are exhibited, separated from each other, there will be no fear
of contusion of the rings, but a sheet of bright light (from the
rays polarized in the plane of reflection) will be spread over
the faint rings that we are seeking, and will effectually make
them invisible.

The plan which I have successfully adopted is, to combine a
tourmaline and doubly-refracting prism. By means of the tour-
maline (with axis perpendicular to the plane of reflection) the
brightness of the sheet of light, which would otherwise cover
the rings that we have to examine, is so far diminished, that it
offers no serious obstacle. At the same time the other set of
rings is seen, and serves very well as an object of comparison.

To destroy the reflection at the upper surface of the imposed
lens is a matter of importance. I have used a plano-convex lens
of 5,8 inches focal length with an obtuse-angled prism placed

416 Prorrssor Arry on the Phenomena

upon its plane side, the obtuse angle being over the center of
the lens. A drop of water was placed between them. Though
its refractive index differs sensibly from that of the glass, vet the
reflection at the common surface of the prism and lens is almost
totally destroyed, for the following reason. The surface of the
lens is I suppose very slightly convex, and when the drop of
water is interposed, and the air-bubbles are rubbed out, Newton’s
rings are seen, very large though slightly irregular, with the
black spot in the center. The rings in question are seen through
this black spot, and consequently are not injured by the effects
of reflection. The water seems to have the power of bringing the
lens and prism into closer contact than is otherwise attainable :*
for I am well convinced that no force that could be applied
without injuring them would bring them so near together as to
exhibit the central black.

For the denser medium I have used a diamond with a surface
of about 4 inch in diameter, mounted in a ring: for the use of

which I am indebted to the politeness of William John Broderip,
Esq. Vice President of the Geological Society. When the lens
and prism were placed on this, a small system of rings was seen
perfectly distinct and well formed, the diameter of the fifth ring
not exceeding 4 of the diameter of the surface.

* I may here mention a curious circumstance which occurred to me in the use of
this combination. After leaving the prism, with the lens hanging to its lower surface,
for one or two days, the water contracted itself to a spot (having partly gone off, 1
suppose, by evaporation) of about ¢ inch in diameter, its outline following most accurately
the course of one of the rings (I think the third) even in its deviations from symmetry.
In this state I was not able to move the lens upon the prism, though I applied a force
parallel to the surface of the prism sufficiently great to shiver large splinters from
the lens. On dipping them into water they instantly dropped asunder.

of Newton’s Rings. 417

These rings were examined with the combination of tourmaline
and doubly refracting prism that I have described. When the
angle of incidence was smal], the rings formed by light polarized
perpendicular to the plane of reflection were seen sufficiently
vivid, with black center, accompanied by the other set of rings
which were faint. When the angle of incidence reached the
polarizing angle of the glass, the first set of rings disappeared.
On increasing the angle, the first set of rings was again seen
with center white. In the most favourable state, the first set
of rings was much more faint than the second, but not so faint
that there could be the slightest doubt upon the fact of the
existence of the rings and the whiteness of the center, as I saw
them repeatedly with every change in the arrangement of the
apparatus, and saw a succession of several rings. The white
spot appeared larger than the dark spot in the other set of rings,
but this I imagine is owing merely to the undefined nature of
the spots, and to the circumstance that, in appreciating their
comparative extent, the eye always gives credit to the brightness
for a greater surface than it can properly claim. In respect of
dimensions of corresponding parts, I could see no difference. On
increasing the angle of incidence, the first set of rings again
disappeared, and reappeared in great brilliancy, the center being
now black.

I am willing to think these experiments important, because
they bear immediately upon a part-of Fresnel’s theory which
has always appeared to me most liable to objection, namely the
formule for the extent of vibration in reflected and refracted
rays. On the truth of Fresnel’s general theory as a mere
geometrical representation, namely that light consists of transversal
vibrations, and that polarized light is light in which all the

418 Proressor Airy on the Phenomena

vibrations are perpendicular to the plane of polarization, I shall
say nothing, because I do not think it will be doubted by any
one who is well acquainted with the experiments and has ex-
amined their agreement with calculation. But on the theorems
for intensity in reflected rays, &c., involving points of the
greatest obscurity, and supported only by very forced suppositions,
any one may I think with reason be sceptical. The phenomena
described here and those described in a former paper (On a
remarkable modification, &c.) depend entirely, in theory, upon
the changes of sign of certain quantities which enter imto
Fresnel’s expressions for these intensities. With respect to the
absolute measure of the intensities I can say nothing, except
that the general appearance of the brightness is sufficiently in
accordance with the law. On the whole I think that these
experiments give great probability to the truth of the formula
considered as a general Jaw: and that they establish with
certainty that part of it which implies that, after passing a
certain angle, the direction of the vibration in the reflected
ray (considered with respect to that im the incident ray) is
reversed.

OssERVATORY, G. B. AIRY.
Feb. 4, 1832. 4

POSTSCRIPT.

Since the above account was written I have (with a favour-
able sky) seen the white-centered rings many times, and several
times with a doubly-refracting prism only, unassisted by a
tourmaline. In examining one part of the phenomena I find
that there is a discordance of a most curious kind from what
the strict theory had led me tu expect.

of Newton's Rings. 419

When the light is incident at the polarizing angle of the
glass, the rings, so far as I can see, vanish totally. Though
I have looked several times with the most scrutinizing attention,
I have not been able to see the least trace. If the angle of
incidence is gradually increased till it exceeds the polarizing
angle, the black-centered rings disappear gradually without al-
tering their size (a considerable quantity of light being still re-
flected from the diamond) and white-centered rings of the same
size appear in their place, without any intermediate stage except
a total absence of rings. From the agreement of this with theory
I conclude that the polarization of light at the inner surface
of glass is (to the senses) complete. But at the polarizing angle
of the diamond the case is perfectly different. On increasing
the angle of incidence till it exceeds this angle, the white-
centered rings do not disappear, but the first black ring con-
tracts so as to leave no central white, and becomes itself the
black center. After this there is no material change: I find
however that the black center of the rings produced by light
polarized perpendicular to the plane of reflection is always
(beyond the polarizing angle of the diamond) sensibly larger
than the black center of the rings produced by light polarized
in the plane of reflection.

The nature of this transition from rings of one character to
rings of the opposite character appears to me to be, theoretically,
extremely curious. As the rings do not disappear, it is plain
that if light polarized perpendicular to the plane of incidence
(or whose vibrations are entirely in that plane) is incident at
what is called the maximum polarizing angle of the diamond,
a portion of it is still reflected. Still however on increasing the
angle of incidence the character of the rings is changed: and

Vol. IV. Part Il. 8H

420 Proressor Atry on the Phenomena

this takes place at an angle where (so far as we are entitled
to conclude) there is nothing peculiar in the reflexion from the
glass; and we are therefore compelled to admit that, the incident

is sue fedabok F
vibration being a. sin = (vé- 2), when the angle of incidence is
increased so as to exceed that angle the reflected vibration is

ve Si Fai a Pest
changed from + p.sin = (vt—a) to-—q.sin a (vt—a). A similar
change takes place at the polarizing angle of the glass: but
there, as we have seen, the transition from +p to —q is effected
by passing through 0, or by the entire cessation of reflection
at one angle of incidence; which is not the case at the po-
larizing angle of the diamond. How then is the gradual change

from + p sin <7 (vt—a) to —q.sin = (vt—a) to be explained? I

answer that the phcenomena prove that it follows from a gradual
change of phase, while the coeflicient is not much altered. In
other words (neglecting the trifling alteration in the coefficient)

the quantity + p sin = (vt—x) is changed to — p sin = (vt—2), not
by the disappearance of p, but by the expression assuming the
form p sin [= (vt—2)-6, where @ increases from 0 to z. This
may be popularly explained in the following manner. The
common Newton’s rings, formed between two lenses, are produced
by the interference of the light reflected from the lower surface
of the upper lens with that reflected from the upper surface
of the lower lens. Now if the upper lens be raised a little, or
the lower depressed a little, the rings contract. As the only
immediate effect of depressing the lower lens is to cause the

light reflected from it to describe a longer path, or to have its
phases retarded, it appears that a contraction of the rings. may

of Newton's Rings. 421

be considered as the effect of a retardation in the phase of the
hight reflected from the lower surface. The contraction of the
rings then in passing the polarizing angle of the diamond requires
us to admit that the phase of the reflected light (the incident
light being polarized perpendicular to the plane of the reflexion)
is, On increasing the angle of incidence by a few degrees, retarded
nearly 180°.

The retardation however is not quite 180°. For if it were, the
character of the rings would be exactly changed, so that the pro-
portion of the size of the central black spot to that of the first
white ring would be the same as that of the central white spot
(before the change) to the first black ring. But as the central
black spot formed by rays polarized perpendicular to the plane
of reflexion is distinctly larger than that formed by rays polarized
in the plane of reflexion, it seems that the black ring has not
contracted completely, or that the alteration of phase is not quite
180°. This reasoning it must be confessed is not certain, as the
same thing would be explained by supposing a small alteration
of phase in the light polarized in the plane of reflexion. I may
mention here that in the Newton’s rings formed between two
lenses of the same kind of glass, the central black spot in those
formed by light polarized perpendicular to the plane of reflexion
is larger than in those formed by the light polarized in the plane
of reflexion.

If, while the white-centered rings are under examination, the
tourmaline and doubly-refracting prism are turned round, the
rings become faint, but do not disappear, and are changed into
black-centered rings by the contraction of the rings. This is
exactly similar to what takes place when a lens is placed on a

metallic surface, and it proves that (as in the former paper),
312

422 Proressor Arry on the Phenomena

while the angle of incidence is a few degrees less than the maxi-
mum polarizing angle of the diamond, the phase of light polar-
ized perpendicular to the plane of reflexion is more retarded than
the phase of light polarized in that plane.

I have not found any variation in these results from changing
the position of the plane of reflexion on the diamond surface.

The result of these experiments and reasonings may be thus
stated.

1. When the angle of incidence is less than the maximum
polarizing angle of the diamond, the nature of its reflexion is
similar to that of metallic reflexion: the phase of vibrations in
the plane of reflexion being more retarded than that of vibrations
perpendicular to the plane of reflexion, but perhaps by a smaller
quantity than in reflexion from metals.

2. In the neighbourhood of the polarizing angle, the nature of
the reflexion is different from any that has hitherto been de-
scribed. The vibrations in the plane of reflexion do not vanish,
but on increasing the angle of incidence by three or four degrees
the phase of vibration is gradually retarded by nearly 180°. In
the reflexion of light whose vibrations are perpendicular to the
plane of reflexion there is no striking difference between the ef-
fects of diamond and those of glass.

3. For angles of incidence greater than the polarizing angle,
there is no sensible difference between the effects of diamond
and those of glass.

I may remark that the extent of vibration in the plane of
reflexion may be represented thus (the formula being purely
empirical and given only for illustration). The vibration in the

incident light being asin = (vt— 2) that in the reflected light is

of Newton’s Rings. 423

tan (c’ — 2”)
tan ( +0’)

Bo be g
asin sa (vt—wx) —bacos = (vt—2) ,

where } is always small but never=0, and is perhaps constant.

The conclusions at which I have arrived are at variance with
one of Sir David Brewster’s (Phil. Trans. 1815). Sir David Brew-
ster’s character as an experimental philosopher stands deservedly
so high, and my estimation of his accuracy, (as observed by
myself in the repetition of many of his experiments) is so great,
that I think it necessary to point out distinctly the nature of
this disagreement.

Sir David Brewster states that homogeneous light is com-
pletely polarized by the diamond at the proper angle. I have
made no experiments here with homogeneous light, and I know
that, on account of its extreme faintness however obtained, little
confidence can be placed in results which depend only on the
evanescence of the reflected light. But the phenomena observed
by me are entirely inconsistent with this supposition. If homo-
geneous light were used, then (on this supposition) the bright-
centered rings would disappear and black centered rings would
succeed them as at the polarizing angle of the glass. If white
light were used, the rings in the neighbourhood of the polarizing
angle would be wholly coloured, and on changing the angle the
intensity of the different colours in each ring would alter, but
there would be nothing like contraction. Thus at a certain angle
the brightest part of the red would be at the center of the spot,
and its faintest part would be in the first ring; while for the
blue the places would be reversed: on increasing the angle the
brightest parts of both would be in the first ring. Whereas in
my experiments there was no discoverable alteration in the
colours of the rings, there never was seen a bright red center

424 Proressor Airy on the Phenomena of Newtons Rine's.
. 5

surrounded by a bright blue ring; but the rings, without chang-
ing their character as to colour, diminished steadily till the
central spot was as it were squeezed out. Whether the only
diamond which I have used may possess any peculiarity which
distinguishes it from those used by Sir David Brewster I cannot
say. Meantime I may observe that the singularity in the re-
flexion at the surface of the diamond makes it not improbable
that there may be some singularity in the refraction also, and
renders a more extended inquiry into the laws both of its re-
flexion and of its refraction highly desirable.

OssERvaTory, G. B. AIRY.

Feb. 16, 1832.

Transactions of the Cambridge Phil. Soe.Vol. 4. Pl. 24.

Joalad Neale Soup 042 Surv

=
-
SP
——
— er at *
> ic ea, &
ie — Ps
a = eee»
ow tae
. ho oN
4

XV. Description of a Machine for resolving by In-
spection certain important Forms of Transcendental

Equations.

By Sir J. F. W. HERSCHEL,

MEMBER OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read May 7, 1832.]

(1.) In the course of a conversation with Mr. Babbage on the
subject of applying machinery to the performance of numerical
computations it occurred to me that, seeing the perfection with
which every description of wheelwork and rectilinear or parallel
motion can now be executed, almost any combination of circular
functions involving as well the are itself and its multiples and
submultiples, as their sines, cosines, chords, &c. might be repre-
sented by the motion of a point, or by the difference of motions
of two points, regulated by mechanism, with almost perfect preci-
sion, and that it would therefore need nothing more than to mark
the arrival of such a point at some definite line or circle, or to cause
the machine in some way or other to come to rest when such dif-
ference should attain a given magnitude, or when any other assign-
ed condition should be fulfilled, to obtain a solution of the equation
expressive of that condition ; which solution should be limited only,
in point of exactness, by the precision of the workmanship and
the accuracy attainable in hitting the coincidence and reading off

426 Sir J. F. W. Herscuet on a Machine

the result. This idea (the principle of which, it will be observed,
is entirely independent of all numerical calculations, or of any
application of wheelwork to perform such calculations, and turns
entirely on the modifications which an uniform motion communi-
cated to the primum mobile can be made to receive in passing
through a train of wheels, levers, &c. and on the accuracy with
which circles and straight lines can be graduated and read off) I
mentioned as it occurred, and presently illustrated it by applica-
tion to the well known equation between the excentric and mean
anomalies in the elliptic motion of a planet.

(2.) The combination of movements which then suggested itself
for this purpose, though theoretically correct, was practically liable
to many objections. It happened however that I was at that time
engaged in investigating the elliptic orbits of some of the most
remarkable double stars; and in the course of that enquiry had
continual occasion for the numerical resolution of cases of this
equation in every state of the data. Finding the preliminary
trials requisite for establishing a rapid conveyance of the succes-
sive approximations consume a great deal of time, even more than
the approximations themselves when once effectually entered upon,
I set myself to consider whether some simple contrivance free from
such objections might not be found which would give me by in-
spection at least a first approximation to the solution, and thus
prove of immediate practical utility. After one or two failures
from attempting to unwind a thread from the circumference of a
wheel revolving on a fixed center, and after rejecting as impracti-
cable Newton’s mechanical solution of the problem by the rolling
of a wheel upon a plane, it occurred to me that a wheel will revolve
uniformly and unwind a thread from it with a uniform motion

just as well whether it revolve on its own center as an axis, or be

for resolving Transcendental Equations. 427

carried round, bodily, by the attachment of its center to the end
of a revolving arm, whose length may either be permanent or
adjustable by a slider. Hence arose the following construction
which seems to be as simple as the nature of the problem
admits.

(3.) In figure 1, Cis a horizontal avis at right angles to the
plane of the paper, on which is firmly fixed at right angles, or in
the plane of rotation of the axis, a cross piece, having a bevelled
vroove PQ, in which the’ slider S# moves freely backwards and
forwards until arrested and fastened by a clamping screw. Pro-
jecting forwards from the slider SR, and firmly rivetted into it is
a short pin B, on which, as on an axis, the excentric wheel DEF
is set, and permitted to turn freely until also arrested and clamped.
These clampings are both easily performed by means of a male
screw cut on the end of the pin B on which is fitted a female
screw cut in the cross head, or clamping key mn. When this screw
is loosened the slider SR can move in its groove and the wheel
DEF revolve on the pin B, but when tightened these motions
are rendered impossible, and the whole apparatus, cross-piece,
slider, and excentric wheel become part of the axis C, and revolve
with it as one mass. Thus we are enabled to adjust, first, the
distance of the center B from the axis, and secondly, the position
of a given point in the circumference of the excentric wheel with
respect to a horizontal line.

(4.) The axis C also carries an index-arm CH furnished either
with a simple index like a clock’s hand, or if accuracy be re-
quired with a vernier adapted to subdivide the graduations of an
index-circle AH concentric with C, and divided into degrees, &e.
Around the excentric wheel DEF a thread of some fine and in-
extensible material (such as dentist’s silk or very delicate silver
wire) is wound. For this purpose the edge of the excentric wheel

Vol. IV. Part III. 31

428 Sir J. F. W. Herscuet on a Machine

must be turned truly cylindric, leaving at either side a slight ele-
vation like a parapet to prevent the thread from slipping off. In
the front edge of this elevation must be cut a small notch F,
in which the end of the thread is to be carried off the edge of
the wheel, and fastened on a pin P stuck (like the tuning pegs
in the handle of a violin) into the face of the wheel, so as to allow
of lengthening or shortening the thread a little by winding it
more or less on this peg. The other end of the thread where it
leaves the circumference of the wheel at H hangs down vertically,
and has suspended to it a vertical, straight, divided scale MN, the
divisions of which are marked and read off by the intersection of
its fiducial line MKWN with the horizontal straight edge 1KJ at K,
for which reading off a microscope moveable on JJ, and furnished
with a micrometer, might be used should such a degree of pre-
cision be required. The straight edge IJ is itself also graduated
for a purpose to be presently explained, and might in like manner
be read off microscopically.

(5.) The axis C also carries on it a barrel abc, round which
and round a fixed smocth pin « a string makes two or three
coils (embracing both the barrel and pin in each coil), after which
it is attached at both its ends to a weight w, which must be such
as to afford a sufficient tension to the string to produce a smooth
uniform friction on the barrel, and thereby to prevent the axis
from turning without the continual application of a moving power,
and to bring it at once to rest, without jerks, dragging, or recoil,
when the power ceases to act. The power may be applied either
at the circumference of the barrel by the hand, or by a handle
fixed on the axis C, and not represented in the figure. The barrel,
weight, and pin x, are supposed to lie behind the plane of the
index circle 4H, which is that of the paper—the rest of the ap-
paratus before it.

for resolving Transcendental Equations. 429

(6.) Let unity (1) represent the radius of the excentric wheel,
plus that of the thread, and e the length of the interval CB between
the centers of the index circle and excentric—or the excentricity
of the latter. Then if we draw AC parallel to the horizon, and
put w for the angle ACB, the perpendicular BG will = e. sin uw.
Now, were the center of the excentric wheel preserved constantly
on the level of the line AC, and that wheel itself merely made
to revolve uniformly by the rotation of C about B instead of B
about C, the part of the thread which would be wound up on its
circumference by this rotation from the commencement of the motion
would be represented by 1xuw=w. This then would be the quan-.
tity by which the vertical scale would, on that supposition, be
raised above its original position. But in the actual case, not only
is the thread so wound on the wheel, but the center B of the
wheel being raised above AC by e sin wu, carries up with it the
wheel, thread, and scale, all by the same quantity. Therefore
the total elevation of the scale due to both causes acting at once
will be w+e.smzu. If then we put 4 for this elevation, there
will subsist between ~ and A the transcendental relation

u+t+e.smu=A,

which is that of the problem of the excentric and mean anomalies
in the elliptic motion of a planet, the arcs being reckoned from
the aphelion. If we would reckon them from the perihelion, we
have only to wrap the thread the other way round the excentric
wheel, when the equation expressing the relation between 4 and x
becomes

u—e.snu= A,

(7.) It appears from what has been said, that the values of «
being read off on the index circle at H, those of A will be read

812

430 Sir J. Ff. W: Herscuet on a Machine

off on the vertical scale at JZ. The zeros of both readings corre-
spond to the horizontal position of the line CB, which position
having been once well ascertained and verified, may be afterwards
at any time recovered by a small level PS screwed on the cross-
piece PQ, and adjusted accordingly. Both zeros are adjustable—
that of the index circle by the stiff-friction motion of the index-
arm CH, and that of the scale, first, coarsely by loosening the
excentric wheel on its center, and winding or unwinding the thread
on its circumference, secondly, more delicately by screwing or
unscrewing the peg p to which its end is fastened.

(8.) The division of the vertical scale must be into 360 equal
parts or degrees, the whole length from 0 to 360 being that of
the circumference of a circle whose radius is 1. This length may
be determined (better than by any attempt to measure the radius)
by turning the axis C round through one complete revolution, and
noting by temporary marks or dots on the scale the points inter-
sected on the fiducial line, by the horizontal straight edge JJ at the
two extremities of its motion. The interval between these dots
is the value of 360 parts required, and must be subdivided ac-
cordingly. It is evident that whatever be the value of e in the
equation w+e¢.sin «= 4 an increase of 360° in « must correspond
to 360° increase in 4, so that the accuracy of this process is, theo-
retically speaking, independent of the position of B on the slider—
but practically it is preferable, before executing this most essential
of all the preliminary operations, to bring the center of the ex-
centric as nearly as possible to coincidence with the axis C, because
in that position any slight deviation from horizontality in the line
IJ will not influence the result.

(9.) The perfect horizontality of this line is however of ma-
terial import to the correct performance of the machine. It is

for resolving Transcendental Equations. 431

easily verified and secured by a spirit level and screw adjustments
at one or both ends. The straight edge JJ itself should be graduated
into equal parts, corresponding to decimals and centesimals of
the radius (1), to obtain which it is only requisite to measure the
length of 360° on the vertical scale as before obtained, and divide
the result by 27 = 2 x3°14159, &e. It is likewise convenient that
the straight edge should have a screw motion in a horizontal direc-
tion, by which its error of zero may be destroyed without altering
its horizontal position.

(10.) The fiducial line of the vertical scale, or that whose in-
tersection with the straight edge marks the values of 4, ought to
be exactly vertical when the scale hangs freely. This condition
is not indeed essential to correct performance, provided it have
been graduated in its inclined position, but, if satisfied, it greatly
facilitates other essential adjustments, and may therefore be regarded
as one of those which must be gone through. It is very easy, all
that is needed being to make the lower end of the vertical scale
terminate in a narrow tail-piece of lead, which being flexible, we
may by bending it a little one way or other ¢ilé the center of
gravity of the scale, so that the fiducial line shall coincide in
direction with a fine plumb line.

(11.) The values of e may be read off in two ways, either, first,
by a graduation on the slider itself, (in which the fixed part may
serve as a vernier to the moveable one,) or on the straight edge. In
the former case the graduation must be, like that of the straight
edge, into decimals and centesimals of the unit radius determined
as above described, and the zero point must be ascertained as follows.
Set the straight edge IJ and the slider both horizontal by their re-
spective levels, adjust the index hand H to 0°, and bring the center
B of the excentric wheel as nearly as may be over C, the center

432 Sir J. F. W. Herscuer on a Machine

of the horizontal axis; then make very slowly one complete revo-
lution of that axis, noting carefully at its commencement, and end,
and at every 30°, the reading off of the straight edge IJ, where it is
intersected by the fiducial line of the vertical scale, (which should
always be allowed to attain perfect rest free from lateral oscilla-
tions). If the point of intersection be found not to have varied
on the straight edge, it is evident that the coimcidence of B and C
must have been perfect; but if otherwise, its extreme variations
will mark out the diameter of the small circle which B continues
to describe about C. This must be destroyed by shifting the place of
B on the slider, and if needed, by altering the place of the slider
itself on the axis (which may possibly have been originally erro-
neous, so as not to allow of the line described by B in its groove,
passing through C at ali). As soon as this is done, and the in-
variability of the above-mentioned intersection ascertained, a fine
line must be drawn directly across the slider and its groove at
each end, and thus the zero points of the divisions both of the
slider and its verniers at each end are secured. The divisions should
be carried along the whole length of the cross piece and along both
edges of the slider, on one forwards, and on the other backwards,
by which means the machine is equally adapted for positive and
negative values of e. If the cross-piece be long enough, the ma-
chine will of course serve for values of e equal to or greater than
unity as well as less.

(12.) If we would read off e on the horizontal straight edge, we
must first set the index hand to 0, and then to 180°—the differ-
ence of the readings is equal to 2e, being the diameter of the
circle described by B transferred to the straight edge by perpen-
diculars to the horizon.

(13.) The only adjustment of any degree of delicacy is that of

Sor resolving Transcendental Equations. 433

the horizontal situation of BC, or of the cross-piece level, the
line BC being imaginary and intangible. Good workmanship will
of course ensure a very near approach to parallelism between this
‘line and the two sides of the cross-piece ; but if an error be still
supposed to exist it may be detected by the following process :
make e =1, or set the slider to 1:00, and then beginning at 0° of
the mdex read off the value of .4, corresponding to equal small
increments of « (for example from degree to degree) round a com-
plete semicircle to 180°. Then, since we have
a = 1+ cos uw, and = = — sin u,

a comparison of differences and second differences will readily
enable us to perceive whether any appreciable deviation from
the true position exists. It is only by the differences of read-
ings that an error in the zero of u can be separated from one in
that of 4, (in which is included that of the index reading at HZ),
as the form of the equation uv +e.sin «= A will easily make
evident.

(14.) However truly the cylindrical form of the excentric wheel
is attained, if the axis of the cylinder be not parallel to that of
the index circle, the thread will wind off an ellipse in place of a
circular are, of which equal portions will not correspond to equal
angles of rotation. his then affords a means of detecting and
rectifying such want of parallelism in the axes in question ; but
as its effect would be confounded with that arising from the error
in the origin of ~ just mentioned in the last article, it will be pre-
ferable to examine through a whole revolution of the index axis
(which should be perfectly horizontal) whether the thread main-
tains precisely the same distance from either edge of the excentric
wheel as it wraps itself round it.

434 Sir J. F. W. Herscuet on a Machine

(15.) Supposing all these adjustments well made, the workman-
ship good, and the graduations such as may be executed, there
seems no reason in the nature of the case why the mechanical
solution afforded by the apparatus we have above described should
not possess equal precision with any astronomical observation, and
therefore be available in many instances where extreme nicety of
computation is not required, or where several hypothetical ellipses
may require to be tried in the calculation of the orbit of a comet
or other celestial body. In the enquiries to which I have already
alluded I found in fact a very material saving of time and trouble
from the use of such an instrument, though constructed in the
rudest manner from materials casually at hand.

(16.) The solution of the equation « + e.sinw = 4 includes
that of its derivative forms

u+a.sm(u+6)=4,
and wu +a.sinu+6.cosu = 4A,

where however only sines and cosines of «~ are involved. If we
would introduce tangents, secants, &c. we must have recourse to
a modification of our mechanism. For instance, suppose the equa-
tion to be resolved were
u+p.tanu= A,

then, retaining the slider, excentric wheel, and horizontal axis C, as
in the contrivance already described, let the straight edge JJ (fig. 2)
instead of being permanently fixed in a horizontal position, be made
to revolve on a center J vertically below C, with half the angular
velocity of the index hand, which may be done either by a toothed
wheel working into another of twice the diameter, or by catgut
wrapping tightly round cylinders in the same proportion, and let
the zero of both rotations be in the horizontal positions of CB and

Sor resolving Transcendental Equations. 435

IJ. Moreover, let the index circle instead of being as before con-
centric with the axis C be now described about the center J, and
the index arm with its vernier be connected with that axis as
we before supposed to be with the axis C.

17.) Put a for the arc read off on this circle or the angle XTJ,
also let e = CB, BD = BE = 1; then since the angle GCB is twice
ATLIJ, or = 2u, we have

MJ = DE + EJ — DEM = DE + BG + KJ — DEM,
=2u+e.sn2u+ KL.tanu— DEM,

DEM is the part of the thread included between its point of at-
tachment to the wheel, and the zero of the scale—it is therefore
an arbitrary constant. If we put c to represent it, and observe
that KL = CL — CG —- GK=CT.cotanu — CB. cos2u-1

=6.cotan u —e.cos2u—1, (putting 4 for the constant distance CZ)

we shall have for the expression of MJ,
MJ =2u+e.sin2u—c+{b.cotan u—e.cos2u-1}.tanw
=(2u+6b-c)+(e—2).tanu.
If therefore we put MJ=2Ad +6-c; and take e=2p+1, the
relation between A and w becomes
u+p.tanu=A,
which is the equation proposed.

(18.) In order then to adopt the above construction to any given
case of this equation we must set the slider so that the distance
BC shall =2p+1. That is to say, the zero of the scale of the
divisions on the slider must commence at a distance from C equal
to the unit or radius of the excentric + that of the thread, and
its graduation must be into parts double of the corresponding parts
of p. In like manner the graduation of the vertical scale MJ must
begin at the point where the revolving straight edge intersects

Vol. IV. Part III. 3K

436 Sir J. F. W. Herscuet on a Machine

it in its horizontal position (in which also CB must be adjusted
to be aceurately horizontal.) and the parts of this scale must be
double of the corresponding parts of A, that is, double degrees of u.
Each part therefore must be one 180th part of the circumference of
the circle DEF, as measured by the unwinding of the thread in
the mode explained in (Art. 8.)

(19.) If the equation proposed were the rather more general one

u+p.tanu+q.secu=A,

we might employ the same construction slightly modified by making
the straight edge IJ revolve attached to an arm at right angles
to the axis of the lower wheel, so that the motion of rotation of IJ
shall as before be uniform, and half as swift as that of CB, but
its direction not passing through the center Q of its revolutions.
All other things remaining as in (Art. 16.), let the perpendicular
QI =f then will the value of MJ be as before,
MJ =2u+esin 2u + (CL - CG — GK.) . tan wu,
but in this case we have
CL =/.cosecu + 6. cotan wu,
so that the value of MJ becomes
MJ=2u+e.sin2u+{f.cosecu+b.cotanu+e.cos2u-1}.tanw,
=(2u+b)+(e-1).tanu+/. sec u,
that is, putting MJ =24+6); e=2p+1; f=24q,
A=u+p.tanut+q.secu.

(20.) It is needless to enlarge on the adjustment of these con-
trivances, which are merely introduced as specimens of the varia-
tions which a trifling change in the construction of our mechanism
is capable of making in the form of the equations resolved. I
will only observe here, that as ovals can now be turned of almost

Sor resolving Transcendental Equations. 437

any excentricity, which shall scarcely deviate perceptibly from the
true elliptic figure, so, in all these and similar cases, by substituting
elliptic for circular excentric wheels, the arc u instead of being
directly related to the sines, cosines, tangents, &c., involved in the
equations may, without the slightest increase of mechanism, or
any additional difficulty in the process of solution, be replaced
by a transcendent of that form which depends on the rectifica-
tion of the ellipse.

(21.) It is almost needless to mention that any mechanical con-
trivance which converts a uniform motion w into another not uni-
form, but varying according to any function ¢ (x) of the former,
affords either a solution of the equation ¢ (uw) = 4, or a tabulation
of the values of ¢(w), just as we please. In the one case we have
only to arrest the motion at equal intervals of the scale on which
the graduation of 4 is engraved, and read off the graduation of
that on which w is represented. In the latter the movements
must be arrested at equal intervals of ~, and the values of 4 read
off. Thus the tabulations of the direct and inverse functions pre-
ceed, pari passu.

(22.) It is not my intention in this paper to enter at large
into the general question of the representation of analytical functions
by continued motion, though perhaps I may take a future oppor-
tunity of so doing. I will only here consider one other case, by
which, without any great complication of machinery, the principle
[ have above adopted may be extended to equations containing
several transcendental relations, such as

u+p.smmu+q.sinnu= A,
and others of the same nature. Suppose, instead of attaching our
excentric wheel to a point in the revolving arm BC we attach it
to a second revolving arm, whose center of rotation occupies the

3kK2

438 Sir J. F. W. Herscuet on a Machine

point which that of the excentric itself occupied in the original
construction, and let this second arm have a yelocity of rotation
in a constant ratio to that of the first, a condition attainable by
contrivances to be presently considered. In that case our construc-
tion will be as in fig. (4), respecting which figure we will establish
the following notation.

CB=e; BB =e; BE'=1; angle ACB=au; DBB = Bu;
MK =x,and DE'M =c,
when we have

MK=«=DEK- DEM=DE+BG- DEM,
=(a+P)u+ BT + BV -c,

=(a+ B)u-—c+e.sinau+e.sin(a+P)u,
assume therefore a=m; a+B=n; e= np; €=nq; r=nA —¢,
and the relation between « and A will be that proposed, viz

ut+p.snmu+q.sinnu= A.

This equation, it will be observed, can always be so prepared as
to make m and x integers, or, if we prefer it, fractions, whose de-
nominators are integers. The form however which will require

the least apparatus of wheels, and into which it is easily trans-
formed, is the following:

A=ut+p.snu+q.sinnu,

in which x is less than unity; for in this state of the equation
the first mover may be applied at once to the axis C, which, as
in the former construction, may carry an index arm reading off «
in degrees on a circle concentric with it. The whole difficulty
then is reduced to the solution of a mechanical problem. To com-
municate to the arm BBP revolving on a center B, attached to an-
other revolving arm CB, a rotation haying a given ratio of velocity

for resolving Transcendental Equations. 439

to that with which the latter arm itself revolves, it being under-
stood that the point of attachment of the center B is to be capable of
adjustment to a greater or less distance from C; a condition which
excludes the use of toothed wheel work. The following is the
simplest construction which has occurred to me for accomplishing
this purpose.

(23.) The horizontal index axis C is surrounded by a grooved
wheel gi, (figs. 4, 5,) which lies behind the index plate (not re-
presented in those figures) and the cross-piece and slider, and is
not attached to the axis so as to turn with it, but on the con-
trary is fixed to the index plate by screws so as to prevent all
rotation. The end / of the cross-piece is penetrated by an axis f
seen in projection in fig. 5, but lengthwise in fig. 6, whose ex-
tremities (to avoid shake and loosening) are pivoted in a bifurcated
and recurved prolongation of the cross-piece, which is seen in
fig. 6 atef. This axis carries on it two wheels also grooved, ed, cd,
both firmly united to the axis, and therefore incapable of moving
unless together as one wheel. A string or band passes round the
groove in gh and cd, so that when the axis C is made to revolve,
and therefore the wheel ed is carried round gf, the latter remain-
ing immoveable, the relative rotation of the one wheel about the
other will wrap and unwrap the string round the groove g/, and
thus produce a rotation of the wheel cd on its axis f, just as it
would do if all the rest of the apparatus stood still, and the wheel
gh alone was turned the other way round the axis C. Thus a
rotation equal and contrary to that of C, or, if the wheels gh, ed,
be of unequal size, in any constant ratio to the latter rotation, is
communicated to the wheels ed, cd’. From the latter of these
let a string be led round a groove in the wheel ab, whose axis
is the pin B on which the second slider revolves, and which

440 Sir J. F. W. Herscuer on a Machine for resolving, &c.

slider is firmly screwed on the face of the wheel, so as to revolve
with it. The distance between the axes f and B of these wheels
being changeable a re-entering string cannot be used, but it must,
after making a complete circumvolution of both, be fastened off
at each of its extremities to pegs, and brought by them into the
requisite state of tension. The rotation of cd will thus be trans-
ferred to ab, increased or diminished in any ratio according to the
ratio of the diameters of the wheels cd and ab, which must be
adjusted accordingly. It is evident that this construction accom-
plishes the end in question, which might indeed be accomplished
without the use of the wheels cd, cd, were it not necessary to
allow the center B completely to attain and even pass across C.
It is very likely that other and still simpler modes of accomplishing
the same object will suggest themselves to others:

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